A Projective-to-Conformal Fefferman-Type Construction

We study a Fefferman-type construction based on the inclusion of Lie groups SL(n+1) into Spin(n+1,n+1). The construction associates a split-signature (n,n)-conformal spin structure to a projective structure of dimension n. We prove the existence of a canonical pure twistor spinor and a light-like co...

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Дата:	2017
Автори:	Hammerl, M., Sagerschnig, K., Šilhan, J., Taghavi-Chabert, A., Zádník, V.
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Мова:	English
Опубліковано:	Інститут математики НАН України 2017
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
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spelling	irk-123456789-1492722019-02-20T01:23:44Z A Projective-to-Conformal Fefferman-Type Construction Hammerl, M. Sagerschnig, K. Šilhan, J. Taghavi-Chabert, A. Zádník, V. We study a Fefferman-type construction based on the inclusion of Lie groups SL(n+1) into Spin(n+1,n+1). The construction associates a split-signature (n,n)-conformal spin structure to a projective structure of dimension n. We prove the existence of a canonical pure twistor spinor and a light-like conformal Killing field on the constructed conformal space. We obtain a complete characterisation of the constructed conformal spaces in terms of these solutions to overdetermined equations and an integrability condition on the Weyl curvature. The Fefferman-type construction presented here can be understood as an alternative approach to study a conformal version of classical Patterson-Walker metrics as discussed in recent works by Dunajski-Tod and by the authors. The present work therefore gives a complete exposition of conformal Patterson-Walker metrics from the viewpoint of parabolic geometry. 2017 Article A Projective-to-Conformal Fefferman-Type Construction / M. Hammerl, K. Sagerschnig, J. Šilhan, A. Taghavi-Chabert, V. Zádník// Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 30 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 53A20; 53A30; 53B30; 53C07 DOI:10.3842/SIGMA.2017.081 http://dspace.nbuv.gov.ua/handle/123456789/149272 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We study a Fefferman-type construction based on the inclusion of Lie groups SL(n+1) into Spin(n+1,n+1). The construction associates a split-signature (n,n)-conformal spin structure to a projective structure of dimension n. We prove the existence of a canonical pure twistor spinor and a light-like conformal Killing field on the constructed conformal space. We obtain a complete characterisation of the constructed conformal spaces in terms of these solutions to overdetermined equations and an integrability condition on the Weyl curvature. The Fefferman-type construction presented here can be understood as an alternative approach to study a conformal version of classical Patterson-Walker metrics as discussed in recent works by Dunajski-Tod and by the authors. The present work therefore gives a complete exposition of conformal Patterson-Walker metrics from the viewpoint of parabolic geometry.
format	Article
author	Hammerl, M. Sagerschnig, K. Šilhan, J. Taghavi-Chabert, A. Zádník, V.
spellingShingle	Hammerl, M. Sagerschnig, K. Šilhan, J. Taghavi-Chabert, A. Zádník, V. A Projective-to-Conformal Fefferman-Type Construction Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Hammerl, M. Sagerschnig, K. Šilhan, J. Taghavi-Chabert, A. Zádník, V.
author_sort	Hammerl, M.
title	A Projective-to-Conformal Fefferman-Type Construction
title_short	A Projective-to-Conformal Fefferman-Type Construction
title_full	A Projective-to-Conformal Fefferman-Type Construction
title_fullStr	A Projective-to-Conformal Fefferman-Type Construction
title_full_unstemmed	A Projective-to-Conformal Fefferman-Type Construction
title_sort	projective-to-conformal fefferman-type construction
publisher	Інститут математики НАН України
publishDate	2017
url	http://dspace.nbuv.gov.ua/handle/123456789/149272
series	Symmetry, Integrability and Geometry: Methods and Applications
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first_indexed	2025-07-12T21:12:56Z
last_indexed	2025-07-12T21:12:56Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 081, 33 pages A Projective-to-Conformal Fefferman-Type Construction Matthias HAMMERL † 1 , Katja SAGERSCHNIG †2, Josef ŠILHAN †3, Arman TAGHAVI-CHABERT †4 and Vojtěch ŽÁDNÍK †5 †1 University of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz 1, 1010 Vienna, Austria E-mail: matthias.hammerl@univie.ac.at †2 INdAM-Politecnico di Torino, Dipartimento di Scienze Matematiche, Corso Duca degli Abruzzi 24, 10129 Torino, Italy E-mail: katja.sagerschnig@univie.ac.at †3 Masaryk University, Faculty of Science, Kotlářská 2, 61137 Brno, Czech Republic E-mail: silhan@math.muni.cz †4 Università di Torino, Dipartimento di Matematica “G. Peano”, Via Carlo Alberto 10, 10123 Torino, Italy E-mail: ataghavi@unito.it †5 Masaryk University, Faculty of Education, Poř́ıč́ı 31, 60300 Brno, Czech Republic E-mail: zadnik@mail.muni.cz Received February 09, 2017, in final form October 09, 2017; Published online October 21, 2017 https://doi.org/10.3842/SIGMA.2017.081 Abstract. We study a Fefferman-type construction based on the inclusion of Lie groups SL(n + 1) into Spin(n + 1, n + 1). The construction associates a split-signature (n, n)- conformal spin structure to a projective structure of dimension n. We prove the existence of a canonical pure twistor spinor and a light-like conformal Killing field on the constructed conformal space. We obtain a complete characterisation of the constructed conformal spaces in terms of these solutions to overdetermined equations and an integrability condition on the Weyl curvature. The Fefferman-type construction presented here can be understood as an alternative approach to study a conformal version of classical Patterson–Walker metrics as discussed in recent works by Dunajski–Tod and by the authors. The present work therefore gives a complete exposition of conformal Patterson–Walker metrics from the viewpoint of parabolic geometry. Key words: parabolic geometry; projective structure; conformal structure; Cartan connec- tion; Fefferman spaces; twistor spinors 2010 Mathematics Subject Classification: 53A20; 53A30; 53B30; 53C07 1 Introduction In conformal geometry the geometric structure is given by an equivalence class of pseudo- Riemannian metrics: two metrics g and ĝ are considered to be equivalent if they differ by a positive smooth rescaling, ĝ = e2fg. In projective geometry the geometric structure is given by an equivalence class of torsion-free affine connections: two connections D and D̂ are considered as equivalent if they share the same geodesics (as unparametrised curves). While conformal and projective structures both determine a corresponding class of affine connections, neither of them induces a single distinguished connection on the tangent bundle. Instead, both structures have canonically associated Cartan connections that govern the respective geometries and encode mailto:matthias.hammerl@univie.ac.at mailto:katja.sagerschnig@univie.ac.at mailto:silhan@math.muni.cz mailto:ataghavi@unito.it mailto:zadnik@mail.muni.cz https://doi.org/10.3842/SIGMA.2017.081 2 M. Hammerl, K. Sagerschnig, J. Šilhan, A. Taghavi-Chabert and V. Žádńık prolonged geometric data of the respective structures. It is therefore often useful when studying projective and conformal structures to work in the framework of Cartan geometries. The present paper investigates a geometric construction that produces a conformal class of split-signature metrics on a 2n-dimensional manifold arising naturally from a projective class of connections on an n-dimensional manifold. Split-signature conformal structures of this type have appeared in several places in the literature before. The projective-to-conformal construc- tion studied in this paper should be understood as a generalisation of the classical Riemann extensions of affine spaces by E.M. Patterson and A.G. Walker [26]. One of the main authors motivations for the present study was the article [15] by M. Dunajski and P. Tod, where the Patterson–Walker construction was generalised to a projectively invariant setting in dimension n = 2. On the other hand, in [25] conformal structures of signature (2, 2) were constructed using Cartan connections that contain the conformal structures arising from 2-dimensional pro- jective structures as a special case. A generalisation of this Cartan-geometric approach to higher dimensions can be found in [24]. In this paper the construction is studied as an instance of a Fefferman-type construction, as formalised in [6, 11], based on an inclusion of the respective Cartan structure groups SL(n+1) ↪→ Spin(n+ 1, n+ 1). We show that in the general situation n ≥ 3 the induced conformal Cartan geometry is non-normal. To obtain information on the conformal structure it is thus important to understand how the normal conformal Cartan connection differs from the induced one, and the main part of the paper concerns the study of this modification. We may summarise the main contributions of the paper as follows: • A comprehensive treatment of the projective-to-conformal Fefferman-type construction including a discussion of the intermediate Lagrangean contact structure (Section 3) and a comparison with Patterson–Walker metrics (Section 6.1). • A thorough study of the normalisation process (Section 4) and an explicit formula for the modification needed to obtain the normal conformal Cartan connection (Section 5.2). • The characterisation of the conformal structures obtained via our Fefferman-type con- struction (culminating in Theorem 4.14). Let us comment upon the characterisation in more detail. This is formulated in terms of a conformal Killing field k and a twistor spinor χ on the conformal space together with a (con- formally invariant) integrability curvature condition. In Theorem 4.14 the properties of k and χ are specified in terms of corresponding conformal tractors, which nicely reflects the algebraic setup of the Fefferman-type construction in geometric terms. An alternative equivalent characterisation theorem was obtained by the authors in [20, Theo- rem 1] by different means, namely, by direct computations based on spin calculus in the spirit of [28, 29]. The conformal properties are given purely in underlying terms and do not refer to tractors. In Section 6.2 (Theorem 6.3) we indicate how this alternative characterisation can be obtained in the current framework. We remark that, to our knowledge, the present work is the first comprehensive treatment of a non-normal Fefferman-type construction and we expect that the techniques developed should have considerable scope for applications to other similar constructions. A particularly interes- ting case of this sort is the Fefferman construction for (non-integrable) almost CR-structures. Possible further applications concern relations between solutions of so-called BGG-equations and special properties of the induced conformal structures. Several such relationships were already obtained by the authors in [20]. For instance, we can give a full description of Einstein metrics contained in the resulting conformal class in terms of the initial projective structure. Moreover, in [21] we were able to show that the obstruction tensor of the induced conformal structure vanishes. A Projective-to-Conformal Fefferman-Type Construction 3 2 Projective and conformal parabolic geometries The standard reference for the background material on Cartan and parabolic geometries pre- sented here is [11]. 2.1 Cartan and parabolic geometries Let G be a Lie group with Lie algebra g and P ⊆ G a closed subgroup with Lie algebra p. A Cartan geometry (G, ω) of type (G,P ) over a smooth manifold M consists of a P -principal bundle G →M together with a Cartan connection ω ∈ Ω1(G, g). The canonical principal bundle G→ G/P endowed with the Maurer–Cartan form constitutes the homogeneous model for Cartan geometries of type (G,P ). The curvature of a Cartan connection ω is the 2-form K ∈ Ω2(G, g), K(ξ, η) := dω(ξ, η) + [ω(ξ), ω(η)], for all ξ, η ∈ X(G), which is equivalently encoded in the P -equivariant curvature function κ : G → Λ2(g/p)∗ ⊗ g, κ(u)(X + p, Y + p) := K ( ω−1(u)(X), ω−1(u)(Y ) ) . (2.1) The curvature is a complete obstruction to a local equivalence with the homogeneous model. If the image of κ is contained in Λ2(g/p)∗ ⊗ p the Cartan geometry is called torsion-free. A parabolic geometry is a Cartan geometry of type (G,P ), where G is a semi-simple Lie group and P ⊆ G is a parabolic subgroup. A subalgebra p ⊆ g is parabolic if and only if its maximal nilpotent ideal, called nilradical p+, coincides with the orthogonal complement p⊥ of p ⊆ g with respect to the Killing form. In particular, this yields an isomorphism (g/p)∗ ∼= p+ of P -modules. The quotient g0 = p/p+ is called the Levi factor; it is reductive and decomposes into a semi- simple part gss0 = [g0, g0] and the center z(g0). The respective Lie groups are Gss0 ⊆ G0 ⊆ P and P+ ⊆ P so that P = G0 n P+ and P+ = exp(p+). An identification of g0 with a subalgebra in p yields a grading g = g−k ⊕ · · · ⊕ g−1 ⊕ g0 ⊕ g1 ⊕ · · · ⊕ gk, where p+ = g1 ⊕ · · · ⊕ gk. We set g− = g−k⊕· · ·⊕g−1. If k is the depth of the grading the parabolic geometry is called \|k\|-graded. The grading of g induces a grading on Λ2p+⊗g ∼= Λ2(g/p)∗⊗g. A parabolic geometry is called regular if the curvature function κ takes values only in the components of positive homogeneity. In particular, any torsion-free or \|1\|-graded parabolic geometry is regular. Given a g-module V , there is a natural p-equivariant map, the Kostant co-differential, ∂∗ : Λk(g/p)∗ ⊗ V → Λk−1(g/p)∗ ⊗ V, (2.2) defining the Lie algebra homology of p+ with values in V ; see, e.g., [11, Section 3.3.1] for the explicit form. For V = g, this gives rise to a natural normalisation condition: parabolic geometries satisfying ∂∗(κ) = 0 are called normal. The harmonic curvature κH of a normal parabolic geometry is the image of κ under the projection ker ∂∗ → ker ∂∗/ im ∂∗. For regular and normal parabolic geometries, the entire curvature κ is completely determined just by κH . A Weyl structure j : G0↪→G of a parabolic geometry (G, ω) over M is a reduction of the P - principal bundle G →M to the Levi subgroup G0 ⊆ P . The class of all Weyl structures, which are parametrised by one-forms on M , includes a particularly important subclass of exact Weyl structures, which are parametrised by functions on M : For \|1\|-graded parabolic geometries, these correspond to further reductions of G0 → M just to the semi-simple part Gss0 of G0 or, equivalently, to sections of the principal R+-bundle G0/G ss 0 → M . The latter bundle is called the bundle of scales and its sections are the scales. For a Weyl structure j : G0 ↪→ G, the pullback j∗ω = j∗ω− + j∗ω0 + j∗ω+ of the Cartan connection may be decomposed according to g = g− ⊕ g0 ⊕ p+. The g0-part j∗ω0 is a principal connection on the G0-bundle G0 → M ; it induces connections on all associated bundles, which are called (exact) Weyl connections. The p+-part j∗ω+ is the so-called Schouten tensor. 4 M. Hammerl, K. Sagerschnig, J. Šilhan, A. Taghavi-Chabert and V. Žádńık 2.2 Tractor bundles and BGG operators Every Cartan connection ω on G → M naturally extends to a principal connection ω̂ on the G-principal bundle Ĝ := G ×P G → M , which further induces a linear connection ∇V on any associated vector bundle V := G ×P V = Ĝ ×G V for a G-representation V . Bundles and connections arising in this way are called tractor bundles and tractor connections. The tractor connections induced by normal Cartan connections are called normal tractor connections. In particular, for the adjoint representation we obtain the adjoint tractor bundle AM := G ×P g. The canonical projection g → g/p and the identification TM ∼= G ×P (g/p) yield a bundle projection Π: AM → TM ; the inclusion p+ ⊆ g and the identification p+ ∼= (g/p)∗ yield a bundle inclusion T ∗M ↪→ AM . This allows us to interpret the Cartan curvature κ from (2.1) as a 2-form Ω on M with values in AM . The holonomy group of the principal connection ω̂ is by definition the holonomy of the Cartan connection ω, i.e., Hol(ω) := Hol(ω̂) ⊆ G. By the holonomy of a geometric structure we mean the holonomy of the corresponding normal Cartan connection. In [12], and later in a simplified manner in [4], it was shown that for a tractor bundle V = G ×P V one can associate a sequence of differential operators, which are intrinsic to the given parabolic geometry (G, ω), Γ(H0) ΘV0→ Γ(H1) ΘV1→ · · · ΘVn−1→ Γ(Hn). The operators ΘVk are the BGG-operators and they operate between the sections of subquotients Hk = ker ∂∗/im ∂∗ of the bundles of V-valued k-forms, where ∂∗ : ΛkT ∗M ⊗V → Λk−1T ∗M ⊗V denotes the bundle map induced by the Kostant co-differential (2.2). The first BGG-operator ΘV0 : Γ(H0) → Γ(H1) is constructed as follows. The bundle H0 is simply the quotient V/V ′, where V ′ ⊆ V is the subbundle corresponding to the largest P - invariant filtration component in the G-representation V . It turns out, there is a distinguished differential operator that splits the projection Π0 : V → H0, namely, the splitting operator, which is the unique map LV0 : Γ(H0)→ Γ(V) satisfying Π0(LV0 (σ)) = σ, ∂∗(d∇ V LV0 (σ)) = 0, for all σ ∈ Γ(H0). The latter condition allows to define the first BGG-operator by ΘV0 := Π1 ◦ d∇ V ◦ LV0 , where Π1 : ker ∂∗ → Γ(H1). The first BGG-operator defines an overdetermined system of differential equations on σ ∈ Γ(H0), ΘV0 (σ) = 0, which is termed the first BGG-equation. 2.3 Further notations and conventions In order to distinguish various objects related to projective and conformal structures, the symbols referring to conformal data will always be endowed with tildes. To write down explicit formulae, we employ abstract index notation, cf., e.g., [27]. Furthermore, we will use different types of indices for projective and conformal manifolds. E.g., on a projective manifold M we write EA := T ∗M , EA := TM , and multiple indices denote tensor products, as in E B A := T ∗M⊗TM . Indices between squared brackets are skew, as in E[AB] := Λ2T ∗M , and indices between round brackets are symmetric, as in E(AB) := S2TM . Analogously, on a conformal manifold M̃ we write Ẽa := T ∗M̃ , Ẽa := TM̃ etc. By E(w) and Ẽ[w] we denote the density bundle over M and M̃ , respectively. Tensor products with other natural bundles are denoted as EA(w) := EA ⊗ E(w), Ẽ[ab][w] := Ẽ[ab] ⊗ Ẽ[w], and the like. 2.4 Projective structures Let M be a smooth manifold of dimension n ≥ 2. A projective structure on M is given by a class, p, of torsion-free projectively equivalent affine connections: two connections D and D̂ A Projective-to-Conformal Fefferman-Type Construction 5 are projectively equivalent if they have the same geodesics as unparametrised curves. This is the case if and only if there is a one-form ΥA ∈ Γ(EA) such that, for all ξA ∈ Γ ( EA ) , D̂Aξ B = DAξ B + ΥAξ B + ΥP ξ P δ B A . An oriented projective structure (M,p), which is a projective structure p on an oriented manifold M , is equivalently encoded as a normal parabolic geometry of type (G,P ), where G = SL(n + 1) and P = GL+(n) n Rn∗ is the stabiliser of a ray in the standard representa- tion Rn+1. Affine connections from the projective class p are precisely the Weyl connections of the corresponding parabolic geometry. Exact Weyl connections are those D ∈ p which preserve a volume form — these are also known as special affine connections. In particular, a choice of D ∈ p reduces the structure group to G0 = GL+(n), if D is special, the structure group is further reduced to Gss0 = SL(n). For later purposes we now give explicit expressions of the main curvature quantities, cf., e.g., [2, 17]. For D ∈ p, the Schouten tensor is determined by the Ricci curvature of D; if D is special, then the Schouten tensor is PAB = 1 n−1R P PA B , in particular, it is symmetric. The projective Weyl curvature and the Cotton tensor are W C AB D = R C AB D + PADδ C B − PBDδ C A , YCAB = 2D[APB]C . Henceforth, we use a suitable normalisation of densities so that the line bundle associated to the canonical one-dimensional representation of P has projective weight −1. Hence, comparing with the usual notation, the density bundle of projective weight w, denoted by E(w), is just the bundle of ordinary ( −w n+1 ) -densities. As an associated bundle to G → M , E(w) corresponds to the 1-dimensional representation of P given by GL+(n) nRn∗ → R+, (A,X) 7→ det(A)w. (2.3) The projective standard tractor bundle is the tractor bundle associated to the standard rep- resentation of G = SL(n + 1). The projective dual standard tractor bundle is denoted by T ∗, i.e., T ∗ := G ×P Rn+1∗. With respect to a choice of D ∈ p, we write T ∗ = ( EA(1) E(1) ) , ∇T ∗C ( ϕA σ ) = ( DCϕA + PCAσ DCσ − ϕC ) . 2.5 Conformal spin structures and tractor formulas Let M̃ be a smooth manifold of dimension 2n ≥ 4. A conformal structure of signature (n, n) on M̃ is given by a class, c, of conformally equivalent pseudo-Riemannian metrics of signa- ture (n, n): two metrics g and ĝ are conformally equivalent if ĝ = f2g for a nowhere-vanishing smooth function f on M̃ . It may be equivalently described as a reduction of the frame bundle of M̃ to the structure group CO(n, n) = R+ × SO(n, n). An oriented conformal structure of signature (n, n) is a conformal structure of signature (n, n) together with fixed orientations both in time-like and space-like directions, equivalently, a reduction of the frame bundle to the group COo(n, n) = R+ × SOo(n, n), the connected component of the identity. An equivariant lift of such a reduction with respect to the 2-fold covering CSpin(n, n) = R+×Spin(n, n)→ COo(n, n) is referred to as a conformal spin structure ( M̃, c ) of signature (n, n). A conformal spin structure of signature (n, n) is equivalently encoded as a normal parabolic geometry of type ( G̃, P̃ ) , where G̃ = Spin(n + 1, n + 1) and P̃ = CSpin(n, n) n Rn,n∗ is the stabiliser of an isotropic ray in the standard representation Rn+1,n+1. 6 M. Hammerl, K. Sagerschnig, J. Šilhan, A. Taghavi-Chabert and V. Žádńık A general Weyl connection is a torsion-free affine connection D̃ such that D̃g ∈ c for any g ∈ c. If D̃g = 0, i.e., D̃ is the Levi-Civita connection of a metric g ∈ c, it is an exact Weyl connection. A choice of Weyl connection reduces the structure group to G̃0 = CSpin(n, n). If the Weyl connection is exact the structure group is further reduced to G̃ss0 = Spin(n, n). Now we briefly introduce the main curvature quantities of conformal structures, cf., e.g., [16]. For g ∈ c, the Schouten tensor, P̃ = P̃(g) = 1 2n− 2 ( R̃ic(g)− S̃c(g) 2(2n− 1) g ) , is a trace modification of the Ricci curvature R̃ic(g) by a multiple of the scalar curvature S̃c(g); its trace is denoted J̃ = gpqP̃pq. The conformal Weyl curvature and the Cotton tensors are W̃ c ab d = R̃ c ab d − 2δc[aP̃b]d + 2gd[aP̃ c b], Ỹcab = 2D̃[aP̃b]c. As for projective structures, we will employ a suitable parametrisation of densities so that the canonical 1-dimensional representation of P̃ has conformal weight −1. Hence, the density bundle of conformal weight w, denoted as Ẽ[w], is just the bundle of ordinary (−w 2n ) -densities. As an associated bundle to the Cartan bundle G̃ → M̃ , it corresponds to the 1-dimensional representation of P̃ given by (R+ × Spin(n, n)) nR2n∗ → R+, (a,A,Z) 7→ a−w. (2.4) In particular, the conformal structure may be seen as a section of Ẽ(ab)[2], which is called the conformal metric and denoted by gab. The spin bundles corresponding to the irreducible spin representations of Spin(n, n) are de- noted by Σ̃+ and Σ̃−, and Σ̃ = Σ̃+ ⊕ Σ̃−. We employ the weighted conformal gamma matrix γ ∈ Γ ( Ẽa ⊗ ( End Σ̃ ) [1] ) such that γpγq + γqγp = −2gpq. For ξ ∈ X ( M̃ ) and χ ∈ Γ ( Σ̃ ) , the Clifford multiplication of ξ on χ is then written as ξ · χ = ξpγpχ. The conformal standard tractor bundle is the associated bundle T̃ := G̃ × P̃ Rn+1,n+1 with respect to the standard representation. It carries the canonical tractor metric h and the con- formal standard tractor connection ∇̃T̃ , which preserves h. With respect to a metric g ∈ c, we have T̃ = Ẽ[−1] Ẽa[1] Ẽ[1]  , h = 0 0 1 0 g 0 1 0 0  , ∇̃T̃c  ρ ϕa σ  =  D̃cρ− P̃ b c ϕb D̃cϕa + σP̃ca + ρgca D̃cσ − ϕc  . (2.5) The BGG-splitting operator is given by LT̃0 : Γ ( Ẽ[1] ) → Γ ( T̃ ) , σ 7→  1 2n ( −D̃pD̃p − J̃ ) σ D̃aσ σ  . (2.6) The spin tractor bundle is the associated bundle S̃ := G̃ × P̃ ∆n+1,n+1, where ∆n+1,n+1 is the spin representation of G̃ = Spin(n + 1, n + 1). Since we work in even signature, it decomposes into irreducibles ∆n+1,n+1 = ∆n+1,n+1 + ⊕ ∆n+1,n+1 − ; the corresponding bundles are denoted by S̃± = G̃ × P̃ ∆n+1,n+1 ± . Under a choice of g ∈ c, these decompose as S̃± = ( Σ̃∓[− 1 2 ] Σ̃±[ 1 2 ] ) , where Σ̃± A Projective-to-Conformal Fefferman-Type Construction 7 are the natural spin bundles as before. For later use we record the formulas for the Clifford action of T̃ on S̃ and for the spin tractor connections on S̃ = S̃+ ⊕ S̃−, ρ ϕa σ  · (τ χ ) = ( −ϕaγaτ + √ 2ρχ ϕaγ aχ− √ 2στ ) , ∇̃S̃c ( τ χ ) = ( D̃cτ + 1√ 2 P̃cpγ pχ D̃cχ+ 1√ 2 γcτ ) , (2.7) cf. [19]. The BGG-splitting operator of S̃± is L S̃± 0 : Γ ( Σ̃± [ 1 2 ]) → Γ ( S̃± ) , χ 7→ ( 1√ 2n D/ χ χ ) , (2.8) where D/ : Γ ( Σ̃± ) → Γ ( Σ̃∓ ) , D/ := γpD̃p, is the Dirac operator. The first BGG-operator associ- ated to S̃± is the twistor operator ΘS̃0 : Γ ( Σ̃± [ 1 2 ]) → Γ ( Ẽa ⊗ Σ̃± [ 1 2 ]) , χ 7→ D̃aχ+ 1 2nγaD/ χ, cf., e.g., [3]. Elements in the kernel of ΘS̃0 are called twistor spinors. It is well known that ΠS̃0 induces an isomorphism between ∇̃S̃-parallel sections of S̃ with ker ΘS̃0 . The adjoint tractor bundle is the associated bundle AM̃ := G̃× P̃ g̃ with respect to the adjoint representation of G̃ on g̃ = so(n+1, n+1) ∼= Λ2Rn+1,n+1. The standard pairing on AM̃ induced by the Killing form on g̃ is denoted as 〈·, ·〉 : AM̃ × AM̃ → R. Henceforth we identify AM̃ with Λ2T̃ . With respect to a metric g ∈ c, AM̃ =  Ẽa[0] Ẽ[a0a1][2] ∣∣ Ẽ[1] Ẽa[2]  . The standard representation of g̃ on Rn+1,n+1 gives rise to the map • : AM̃ ⊗ T̃ → T̃ ,  ρa µa0a1 ∣∣ ϕ βa  •  ν ωb σ  =  ρrωr − ϕν µb rωr − σρb − νβb βrωr + ϕσ  . (2.9) The normal tractor connection is given by ∇̃AM̃c  ρa µa0a1 ∣∣ ϕ ka  =  D̃cρa − P̃ p c µpa − P̃caϕ( D̃cµa0a1 + 2gc[a0ρa1] +2P̃c[a0ka1] ) ∣∣ (D̃cϕ− P̃ p c kp + ρc ) D̃cka − µca + gcaϕ  . (2.10) Written as a two-form Ω̃ with values in Λ2T̃ , the curvature of ∇̃T̃ is Ω̃c0c1 =  −Ỹac0c1 W̃c0c1a0a1 ∣∣ 0 0  ∈ Γ ( Ẽ[c0c1] ⊗AM̃ ) . (2.11) The BGG-splitting operator LAM̃0 : Γ ( Ẽa ) = Γ ( Ẽa[2] ) → Γ ( AM̃ ) , ka 7→  ρa µa0a1 ∣∣ ϕ ka  , 8 M. Hammerl, K. Sagerschnig, J. Šilhan, A. Taghavi-Chabert and V. Žádńık is determined by µa0a1 = D̃[a0ka1], ϕ = − 1 2n gpqD̃pkq, (2.12) ρa = − 1 4n D̃pD̃pka + 1 4n D̃pD̃akp + 1 4n2 D̃aD̃ pkp + 1 n P̃ p akp − 1 2n J̃ka, and the corresponding first BGG-operator of AM̃ is computed as ΘAM̃0 : Γ ( Ẽa[2] ) → Γ ( Ẽ(ab)0 [2] ) , ξa 7→ D̃(cξa)0 , where the subscript 0 denotes the trace-free part. Thus ΘAM̃0 is the conformal Killing operator and solutions to the first BGG-equation are conformal Killing fields. In a prolonged form, the conformal Killing equation is equivalent to ∇̃AM̃b s = ξaΩ̃ab, (2.13) where s = LAM̃0 (ξ), see [7, 18]. 3 The Fefferman-type construction The construction of split-signature conformal structures from projective structures discussed in this section fits into a general scheme relating parabolic geometries of different types. Namely, it is an instance of the so-called Fefferman-type construction, whose name and general procedure is motivated by Fefferman’s construction of a canonical conformal structure induced by a CR structure, see [6] and [11] for a detailed discussion. 3.1 General procedure Suppose we have two pairs of semi-simple Lie groups and parabolic subgroups, (G,P ) and ( G̃, P̃ ) , and a Lie group homomorphism i : G→ G̃ such that the derivative i′ : g→ g̃ is injective. Assume further that the G-orbit of the origin in G̃/P̃ is open and that the parabolic P ⊆ G contains Q := i−1 ( P̃ ) , the preimage of P̃ ⊆ G̃. Given a parabolic geometry (G →M,ω) of type (G,P ), one first forms the Fefferman space M̃ := G/Q = G ×P P/Q. (3.1) Then ( G → M̃, ω ) is automatically a Cartan geometry of type (G,Q). As a next step, one considers the extended bundle G̃ := G ×Q P̃ with respect to the homomorphism Q→ P̃ . This is a principal bundle over M̃ with structure group P̃ and j : G ↪→ G̃ denotes the natural inclusion. The equivariant extension of ω ∈ Ω1(G, g) yields a unique Cartan connection ω̃ind ∈ Ω1 ( G̃, g̃ ) of type ( G̃, P̃ ) such that j∗ω̃ind = i′ ◦ ω. Altogether, one obtains a functor from parabolic geometries (G →M,ω) of type (G,P ) to parabolic geometries ( G̃ → M̃, ω̃ind ) of type ( G̃, P̃ ) . The relation between the corresponding curvatures is as follows: The previous assumptions yield a linear isomorphism g̃/p̃ ∼= g/q and an obvious projection g/q→ g/p, where q ⊆ p is the Lie algebra of Q ⊆ P . Composing these two maps one obtains a linear projection g̃/p̃ → g/p, whose dual map is denoted as ϕ : (g/p)∗ → (g̃/p̃)∗. Since i′ : g → g̃ is a homomorphism of Lie algebras, the curvature function κ̃ind : G̃ → Λ2(g̃/p̃)∗ ⊗ g̃ is related to κ : G → Λ2(g/p)∗ ⊗ g by κ̃ind ◦ j = (Λ2ϕ⊗ i′) ◦ κ. We note that κ̃ind is fully determined by this formula. Since i′ is an embedding, the notation is in most cases simplified such that we write g ⊆ g̃, q = g ∩ p̃, etc. A Projective-to-Conformal Fefferman-Type Construction 9 3.2 Algebraic setup and the homogeneous model Here we specify the general setup for Fefferman-type constructions from Section 3.1 according to the description of oriented projective and conformal spin structures given in Sections 2.4 and 2.5, respectively. Let Rn+1,n+1 be the real vector space R2n+2 with an inner product, h, of split-signature. Let ∆n+1,n+1 + and ∆n+1,n+1 − be the irreducible spin representations of G̃ := Spin(n+ 1, n+ 1) as in Section 2.5. We fix two pure spinors sF ∈ ∆n+1,n+1 − and sE ∈ ∆n+1,n+1 ± with non-trivial pairing, which is assigned for later use to be 〈sE , sF 〉 = −1 2 . Note that sE lies in ∆n+1,n+1 + if n is even or in ∆n+1,n+1 − if n is odd. Let us denote by E,F ⊆ Rn+1,n+1 the kernels of sE , sF with respect to the Clifford multi- plication, i.e., E := { X ∈ Rn+1,n+1 : X · sE = 0 } , F := { X ∈ Rn+1,n+1 : X · sF = 0 } . The purity of sE and sF means that E and F are maximally isotropic subspaces in Rn+1,n+1. The other assumptions guarantee that E and F are complementary and dual each other via the inner product h. Hence we use the decomposition Rn+1,n+1 = E ⊕ F ∼= Rn+1 ⊕ Rn+1∗ (3.2) to identify the spinor representation ∆n+1,n+1 = ∆n+1,n+1 + ⊕∆n+1,n+1 − with the exterior power algebra Λ•E ∼= Λ•Rn+1, whose irreducible subrepresentations are ∆n+1,n+1 − ∼= ΛevenRn+1 and ∆n+1,n+1 + ∼= ΛoddRn+1. When n is even, respectively, odd, we can identify ( ∆n+1,n+1 − )∗ ∼= ∆n+1,n+1 + , respectively ( ∆n+1,n+1 − )∗ ∼= ∆n+1,n+1 − . Now, let us consider the subgroup in G̃ defined by G := {g ∈ Spin(n+ 1, n+ 1): g · sE = sE , g · sF = sF }. This subgroup preserves the decomposition (3.2) so that the restriction of the action to F is dual to the restriction to E. It further preserves the volume form on E, respectively F ∼= E∗, which is determined by sE and sF according to the previous identifications. Hence G ∼= SL(n+ 1) and this defines an embedding i : SL(n+ 1) ↪→ Spin(n+ 1, n+ 1).1 The G-invariant decomposition (3.2) determines a G-invariant skew-symmetric involution K ∈ so(n+ 1, n+ 1) acting by the identity on E and minus the identity on F . The relationship among K, sE and sF may be expressed as h(X,K(Y )) = −h(K(X), Y ) = 2〈sE , (X ∧ Y ) · sF 〉, (3.3) where (X ∧ Y ) · sF = 1 2 (X · Y · sF − Y ·X · sF ) = X · Y · sF + h(X,Y )sF . The spin action of g̃ is denoted by •, and thus A • s = −1 4A · s, for any A ∈ g̃ and s ∈ ∆. In particular, K •sF = −1 2(n+1)sF and K •sE = 1 2(n+1)sE . Here we identify g̃ = so(n+1, n+1) with Λ2Rn+1,n+1. It is convenient to split g̃ in terms of irreducible g-modules as g̃ = Λ2(E ⊕ F ) = (E ⊗ F )0︸︷︷︸ g=sl(n+1) ⊕ (E ⊗ F )Tr ⊕ Λ2E ⊕ Λ2F︸︷︷︸ g⊥ , (3.4) 1Instead of the embedding SL(n+ 1) ↪→ Spin(n+ 1, n+ 1) we could also consider the embedding SL(n+ 1) ↪→ SO(n+ 1, n+ 1). The advantage of employing the embedding into the spin group is two-fold: on the one hand, it is then seen directly that the induced conformal structure has a canonical spin structure, and, on the other hand, we can then use convenient spinorial objects for its characterisation. 10 M. Hammerl, K. Sagerschnig, J. Šilhan, A. Taghavi-Chabert and V. Žádńık where (E ⊗F )Tr = RK, and K acts as [K,φ] = 2φ, [K,ψ] = −2ψ, [K,λ] = 0, for any φ ∈ Λ2E, ψ ∈ Λ2F and λ ∈ E ⊗ F . Further, the annihilators of sE and sF in g̃ are the subalgebras ker sE = sl(n+ 1)⊕ Λ2E and ker sF = sl(n+ 1)⊕ Λ2F . The homogeneous model for conformal spin structures of signature (n, n) is the space of isotropic rays in Rn+1,n+1, G̃/P̃ ∼= Sn × Sn. The subgroup G ⊆ G̃ does not act transitively on that space. According to the decomposition (3.2), there are three orbits: the set of rays contained in E, the set of rays contained in F , and the set of isotropic rays that are neither contained in E nor in F . Note that only the last orbit is open in G̃/P̃ , which is one of the requirements from Section 3.1. Therefore, we define P̃ ⊆ G̃ to be the stabiliser of a ray through a light-like vector ṽ ∈ Rn+1,n+1 \ (E ∪F ). Denoting by Q = i−1(P̃ ) the stabiliser of the ray R+ṽ in G, we have the identification of G/Q with the open orbit of the origin in G̃/P̃ . The subgroup Q, which is not parabolic, is contained in the parabolic subgroup P ⊆ G defined as the stabiliser in G of the ray through the projection of ṽ to E. In particular, G/P is the standard projective sphere Sn, the homogeneous model of oriented projective structures of dimension n, and G/Q → G/P is the canonical fibration with the standard fibre P/Q, whose total space is the model Fefferman space. Let us denote by L = Rṽ the line spanned by the light-like vector ṽ and let L⊥ be the orthogonal complement in Rn+1,n+1 with respect to h. The tangent space of G/Q at the origin can be seen in three different ways, namely,( L⊥/L ) [1] ∼= g/q ∼= g̃/p̃. The latter isomorphism is induced by the embedding g ⊆ g̃, the former one by the standard action of g ⊆ g̃ on the vector ṽ ∈ Rn+1,n+1. Both these identifications are Q-equivariant. There are several natural Q-invariant objects that in turn yield distinguished geometric ob- jects on the general Fefferman space. The n-dimensional Q-invariant subspace f := (( F̄ + L ) /L ) [1] ⊆ ( L⊥/L ) [1], where F̄ := F ∩ L⊥, which is isomorphic to p/q ⊆ g/q, the kernel of the projection g/q→ g/p. Another n-dimensional Q-invariant subspace is e := (( Ē + L ) /L ) [1] ⊆ ( L⊥/L ) [1], where Ē := E ∩ L⊥. The intersection e ∩ f is 1-dimensional with a distinguished Q-invariant generator that corre- sponds to the G-invariant involution K ∈ g̃, k := K + p̃ ∈ g̃/p̃. Note that all these objects are isotropic with respect to the natural conformal class induced by the restriction of h to L⊥ ⊆ Rn+1.n+1. In particular, both e and f are maximally isotropic subspaces such that k ∈ e ∩ f ⊆ k⊥ = e+ f. (3.5) In Section 3.1 we introduced a map ϕ : (g/p)∗ → (g̃/p̃)∗, the dual map to the projection g̃/p̃ ∼= g/q→ g/p. The kernel of this projection is just f and the image of ϕ is identified with its annihilator, which will be denoted by f◦. Since f is a maximally isotropic subspace in g̃/p̃ ∼= g/q, f◦ ∼= f [−2]. Since (g̃/p̃)∗ ∼= p̃+, we may conclude with the help of explicit matrix realisations from Ap- pendix A that f◦ = p̃+ ∩ ker sF . Moreover, we note that( p̃+ ∩ ker sF ) E⊗F = p+, ( p̃ ∩ ker sF ) E⊗F = p, (3.6) Λ2F ∩ p̃ = Λ2F̄ ⊆ g̃0, [ p̃+,Λ 2F̄ ] = f◦, [ f◦,Λ2F̄ ] = 0. (3.7) A Projective-to-Conformal Fefferman-Type Construction 11 3.3 The Fefferman space and induced structure The pairs of Lie groups (G,P ) and ( G̃, P̃ ) from the previous subsection satisfy all the properties to launch the Fefferman-type construction. Proposition 3.1. The Fefferman-type construction for the pairs of Lie groups (G,P ) and( G̃, P̃ ) yields a natural construction of conformal spin structures ( M̃, c ) of signature (n, n) from n-dimensional oriented projective structures (M,p). The Fefferman space M̃ is identified with the total space of the weighted cotangent bundle without the zero section T ∗M(2)\{0}. Proof. The first part of the statement is obvious from the general setting for Fefferman-type constructions and the Cartan-geometric description of oriented projective and conformal spin structures. The second part is shown due to two natural identifications: On the one hand, the Fefferman space is by (3.1) equal to the total space of the associated bundle M̃ ∼= G ×P P/Q over M . On the other hand, the weighted cotangent bundle to M is identified with the associated bundle T ∗M(2) ∼= G ×P (g/p)∗(2) with respect to action of P induced by the adjoint action and the representation (2.3) for w = 2. Hence it remains to verify that the action of P on (g/p)∗(2)\{0} is transitive and Q is a stabiliser of a non-zero element. But this is a purely algebraic task, which may be easily checked in a concrete matrix realisation. � From the algebraic setup in Section 3.2 we easily conclude number of specific features of the induced conformal structure on M̃ : Proposition 3.2. The conformal spin structure ( M̃, c ) induced from an oriented projective structure (M,p) by the Fefferman-type construction admits the following tractorial objects that are all parallel with respect to the induced tractor connection: (a) pure tractor spinors sE ∈ Γ ( S̃± ) and sF ∈ Γ ( S̃− ) with non-trivial pairing, (b) a tractor endomorphism K ∈ Γ ( AM̃ ) which is an involution, i.e., K2 = idT̃ , and which acts by the identity, respectively minus the identity on the maximally isotropic complemen- tary subbundles Ẽ := ker sE, respectively F̃ := ker sF of T̃ . The corresponding underlying objects η = ΠS̃0 (sE), χ = ΠS̃0 (sF ) and k = ΠAM̃0 (K) satisfy: (c) η ∈ Γ ( Σ̃± [ 1 2 ]) and χ ∈ Γ ( Σ̃− [ 1 2 ]) are pure spinors, whose kernels ẽ := ker η and f̃ := kerχ have 1-dimensional intersection and f̃ coincides with the vertical subbundle of M̃ →M , (d) k ∈ Γ ( TM̃ ) is a nowhere-vanishing light-like vector field generating the intersection ẽ∩ f̃ . Proof. The G-invariant spinor sE ∈ ∆± gives rise to the tractor spinor sE ∈ Γ ( S̃± = G ×Q ∆± ) such that it corresponds to the constant (Q-equivariant) map G → ∆±. Hence sE is automatically parallel with respect to the induced tractor connection on S̃±. Similar reasoning for other G-invariant objects and their compatibility described above yield the first part of the statement. In particular, Ẽ = G ×Q E, F̃ = G ×Q F and the decomposition T̃ = Ẽ ⊕ F̃ corresponds to the decomposition (3.2). The filtration L ⊆ L⊥ ⊆ Rn+1,n+1 gives rise to the filtration of the standard tractor bundle, which can be written asẼ[−1] 0 0  ⊆ Ẽ[−1] Ẽa[1] 0  ⊆ Ẽ[−1] Ẽa[1] Ẽ[1]  = T̃ . 12 M. Hammerl, K. Sagerschnig, J. Šilhan, A. Taghavi-Chabert and V. Žádńık In particular, the subbundles associated to Ē, F̄ ⊆ L⊥ are distinguished by the middle slot. The corresponding Q-invariant maximally isotropic subspaces e, f ⊆ g/q determine the distributions G ×Q e and G ×Q f in TM̃ = G ×Q g/q. According to the tractor Clifford action (2.7) it follows that these are precisely the kernels of the spinors η and χ. Since these subspaces are maximally isotropic, the corresponding spinors are pure. Since f ∼= p/q is the kernel of the projection g/q → g/p, the corresponding subbundle f̃ is identified with the vertical subbundle of the projection M̃ → M . The intersection e ∩ f is 1-dimensional and it is generated by the projection of K ∈ g̃ to g̃/p̃. Indeed, K cannot be contained in p̃, since K acts by the identity on E and minus the identity on F and p̃ is the stabiliser of a line that is neither contained in E nor in F . Altogether, the corresponding vector field k on M̃ is a nowhere-vanishing generator of ẽ ∩ f̃ , in particular, it is light-like. � 3.4 Relating tractors, Weyl structures and scales As a technical preliminary for further study we now relate natural objects associated to the original projective Cartan geometry (G, ω) on M and the induced conformal geometry (G̃, ω̃ind) on the Fefferman space M̃ . Since G ⊆ G̃, any G̃-representation V is also a G-representation, which yields compatible tractor bundles over M and M̃ with compatible tractor connections: V = G ×P V → M with the tractor connection ∇ induced by ω and Ṽ = G̃ × P̃ V = G ×Q V → M̃ with the tractor connection ∇̃ind induced by ω̃ind. Sections of V bijectively correspond to P -equivariant functions ϕ : G → V , while sections of Ṽ correspond to Q-equivariant functions ϕ : G → V . Since Q ⊆ P , every section of V gives rise to a section of Ṽ, and we can view Γ(V) ⊆ Γ ( Ṽ ) . Now, Proposition 3.2 in [8] admits a straightforward generalisation to Fefferman-type constructions for which P/Q is connected and thus, in particular, to the one studied in this article: Proposition 3.3. (a) A section s ∈ Γ ( Ṽ ) is contained in Γ(V) (i.e., the corresponding Q-equivariant function ϕ is indeed P -equivariant) if and only if ∇̃inds is strictly horizontal (i.e., va∇̃ind a s = 0 for all va ∈ Γ ( f̃ ) ). (b) The restriction of ∇̃ind to Γ(V) ⊆ Γ ( Ṽ ) coincides with the tractor connection ∇. Remark 3.4. Another instance of compatible bundles over M and M̃ is provided by the density bundles E(w) and Ẽ[w], which are defined via the representation of P and P̃ as in (2.3) and (2.4), respectively. Restricting these representations to Q, it easily follows that the notation is indeed compatible so that we can view Γ(E(w)) ⊆ Γ ( Ẽ[w] ) . Both projective and conformal density bundles can be described as associated bundles to the respective bundles of scales. Hence everywhere positive sections of density bundles are considered as scales. In particular, the inclusion Γ(E+(1)) ⊆ Γ ( Ẽ+[1] ) may be interpreted so that any projective scale induces a conformal one. Such conformal scales will be called reduced scales. An intrinsic characterisation of reduced scales among all conformal ones is formulated in Proposition 5.2. The previous remark yields that any projective exact Weyl structure on M induces a confor- mal exact Weyl structure on M̃ . This fact can be generalised as follows: Proposition 3.5. Any projective (exact) Weyl structure on M induces a conformal (exact) Weyl structure on the Fefferman space M̃ . Proof. A version of this result in a more general context was proved in [1, Proposition 6.1]: any Weyl structure for ω induces a Weyl structure for ω̃ind if P+ ⊆ P̃ and ( G0 ∩ P̃ ) ⊆ G̃0. But both A Projective-to-Conformal Fefferman-Type Construction 13 these conditions are satisfied as follows from the setup in Section 3.2 and explicit realisations in Appendix A. � Conformal Weyl structures induced by projective ones as above will be called reduced Weyl structures. 3.5 Normality Here we show that our Fefferman-type construction does not preserve the normality in general, see Proposition 3.8. This can be shown directly as we did in a previous version of the article, see arXiv:1510.03337v2. Alternatively, we can treat the construction as the composition of two other constructions via a natural intermediate Lagrangean contact structure. A Lagrangean contact structure on M ′ consists of a contact distribution H ⊆ TM ′ together with a decomposition H = e′⊕f ′ into two subbundles that are maximally isotropic with respect to the Levi form H × H → TM ′/H. Such structure on a manifold M ′ of dimension 2n − 1 is equivalently encoded as a normal parabolic geometry of type (G,P ′), where G = SL(n+ 1) and P ′ ⊆ G is the stabiliser of a flag of type line-hyperplane in the standard representation Rn+1. For n > 2 there are three harmonic curvatures, two of which are torsions whose vanishing is equivalent to the integrability of the respective subbundles e′, f ′ ⊆ H. For n = 2 there are two harmonic curvatures of homogeneity 4, hence the Cartan connection is torsion-free. In that case both e′ and f ′ are 1-dimensional and thus automatically integrable. On the one hand, P ′ is contained in P , where P ⊆ G is the stabiliser of a ray in Rn+1. For suitable choices as in Appendix A, the Lie algebra to P ′ consists of matrices of the form p′ = a U t w 0 B V 0 0 c  . Given a projective Cartan geometry (G →M,ω) of type (G,P ), it turns out that the correspon- dence space M ′ := G/P ′ can be identified with the projectivised cotangent bundle P(T ∗M). The Cartan geometry (G → M ′, ω) of type (G,P ′) is regular and thus it covers a natural La- grangean contact structure on M ′. In particular, the canonical contact distribution on P(T ∗M) coincides with H and the vertical subbundle of the projection M ′ →M coincides with one of the two distinguished subbundles, say f ′ ⊆ H. As in general, this construction preserves normality. In accord with [5], respectively [11, Section 4.4.2] we may state: Proposition 3.6. Let (G → M,ω) be a normal projective parabolic geometry and let (G → M ′, ω) be the corresponding normal Lagrangean contact parabolic geometry. The latter geometry is torsion-free if and only if n = 2 or it is flat, i.e., the initial projective structure is flat. On the other hand, P ′ contains Q, where Q = G ∩ P̃ as before. This allows us to consider the Fefferman-type construction for the pairs (G,P ′) and (G̃, P̃ ). Given a Lagrangean contact structure on M ′, it induces a conformal spin structure on M̃ = G/Q. This construction is indeed very similar to the original Fefferman construction; one deals with different real forms of the same complex Lie groups in the two cases. That is why the following statement and its proof is analogous to the one for the CR case. Following [8], respectively [11, Section 4.5.2] we may state: Proposition 3.7. Let (G →M ′, ω) be the normal Lagrangean contact parabolic geometry and let( G̃ → M̃, ω̃ind ) be the conformal parabolic geometry obtained by the Fefferman-type construction. Then ω̃ind is normal if and only if ω is torsion-free. https://arxiv.org/abs/1510.03337v2 14 M. Hammerl, K. Sagerschnig, J. Šilhan, A. Taghavi-Chabert and V. Žádńık Altogether, composing the two previous steps we obtain our projective-to-conformal Feffer- man-type construction with the desired control of the normality. Note that from (3.5) and the respective matrix realisations it follows that the induced objects on M̃ = T ∗M(2) \ {0} from Proposition 3.2 correspond to the induced objects on M ′ = P(T ∗M). In particular, the vertical subbundle of the projection M̃ →M ′ is spanned by k and the decomposition k⊥ = ẽ⊕ f̃ ⊆ TM̃ descends to the decomposition H = e′ ⊕ f ′ ⊆ TM ′ G̃ P̃ �� G * 77 Q **P ′ �� P �� M̃ yy M ′ xx M. Proposition 3.8. Let (G → M,ω) be a normal projective parabolic geometry and let ( G̃ → M̃, ω̃ind ) be the conformal parabolic geometry obtained by the Fefferman-type construction. (a) If dim M = 2 then ω̃ind is normal. (b) If dim M > 2 then ω̃ind is normal if and only if ω is flat. Moreover, independently of the dimension of M , ω̃ind is flat if and only if ω is flat. 3.6 Remarks on torsion-free Lagrangean contact structures At this stage it is easy to formulate a local characterisation of split-signature conformal structures arising from torsion-free Lagrangean contact structures, see Proposition 3.10. As before, the results and their proofs are very analogous to those in the CR case, therefore we just quickly indicate the reasoning and point to differences. As in Proposition 3.2, the G-invariant algebraic objects induce the tractor fields sE , sF and K on the conformal Fefferman space that are parallel with respect to the induced tractor connection and have the required compatibility properties. But, starting with a torsion-free Lagrangean contact structure, the induced connection is already normal. In particular, the corresponding underlying objects χ, η and k are pure twistor spinors and a light-like conformal Killing field, respectively. The existence of parallel tractors sE , sF and K with the algebraic properties as in Proposi- tion 3.2 are by no means independent conditions: Proposition 3.9. Let ( M̃, c ) be a conformal spin structure of split-signature (n, n). Then the following conditions are locally equivalent: (a) The spin tractor bundle admits two pure parallel tractor spinors sE ∈ Γ ( S̃± ) and sF ∈ Γ ( S̃− ) with non-trivial pairing. (b) The conformal holonomy Hol(c) reduces to SL(n + 1) ⊆ Spin(n + 1, n + 1) preserving a decomposition into maximally isotropic subspaces E ⊕ F = Rn+1,n+1. (c) The adjoint tractor bundle admits a parallel involution K ∈ Γ ( AM̃ ) , i.e., K2 = idT̃ . A Projective-to-Conformal Fefferman-Type Construction 15 The only subtle point within the proof concerns the consequences of property (c). The existence of a parallel skew-symmetric involution K on the standard tractor bundle immediately implies that the conformal holonomy Hol(c) is reduced to GL(n + 1). But, analogously to the corresponding discussion for the CR case in [9] or [23], one can show that Hol(c) is actually contained in SL(n+ 1). The rest follows easily. It turns out that conformal spin structures induced by torsion-free Lagrangean contact struc- tures are locally characterised by any of the three equivalent conditions above. Indeed, according to results from [10], the holonomy reduction of the conformal structure to G = SL(n + 1) ⊆ Spin(n+ 1, n+ 1) = G̃ yields the so-called curved orbit decomposition of M̃ , which corresponds to the decomposition of the homogeneous model G̃/P̃ with respect to the action of G. Each subset from the decomposition of M̃ , provided it is non-empty, further carries a geometry of the same type as its counterpart in the homogeneous model. From Section 3.2 we know there is one open and two closed n-dimensional orbits. The closed n-dimensional orbits carry Cartan geometries of type (G,P ), and thus inherit projective structures, the open orbit carries a Cartan geometry of type (G,Q). Note that the two closed orbits coincide with the zero sets of χ and η, the open subset is the one where both spinors, and thus k, are non-vanishing. Since k is the conformal Killing field corresponding to the parallel adjoint tractor K, it inserts trivially into the curvature of the normal Cartan connection, cf. (2.13). Hence, according to [5], the Cartan geometry of type (G,Q) on the open orbit of M̃ descends to a Cartan geometry of type (G,P ′) on the local leaf space M ′ determined by k. It follows that this Cartan geometry is torsion-free and thus determines a torsion-free Lagrangean contact structure. Altogether, following [9] we may state the following characterisation: Proposition 3.10. A split-signature conformal spin structure is locally induced by a torsion- free Lagrangean contact structure via the Fefferman-type construction if and only if any of the equivalent conditions from Proposition 3.9 holds and the underlying twistor spinors χ and η and the conformal Killing field k are nowhere-vanishing. 3.7 The exceptional case: dimension n = 2 From Section 3.5 we know that the intermediate 3-dimensional Lagrangean contact structure on M ′ induced by a 2-dimensional projective structure on M is torsion-free. Hence the induced conformal Cartan geometry on M̃ is normal and thus all the equivalent conditions from Propo- sition 3.9 are satisfied. Moreover, the fact that it comes from a projective structure implies that any vertical vector of the projection M̃ →M inserts trivially into the Cartan curvature, i.e., iX κ̃(u) = 0, for all X ∈ f , u ∈ G̃. (3.8) Analogously to the discussion before Proposition 3.10 we may conclude: Proposition 3.11. A conformal spin structure of signature (2, 2) is locally induced by a 2- dimensional projective structure via the Fefferman-type construction if and only if any of the equivalent conditions from Proposition 3.9 holds, the underlying twistor spinors χ and η and the conformal Killing field k are nowhere-vanishing and the curvature of the normal conformal Cartan connection satisfies (3.8). Remark 3.12. Conformal structures induced from 2-dimensional projective structures are well- studied, see, e.g., [14, 15, 25]. Notably, the intermediate 3-dimensional Lagrangean contact structure can be equivalently viewed as a path geometry (or the geometry associated to second order ODEs modulo point transformations). Such structure is induced by a projective structure (i.e., the paths are the unparametrised geodesics of the projective class of connections) if and only if one of the two harmonic curvatures vanishes. It follows from [25] that this is equivalent to 16 M. Hammerl, K. Sagerschnig, J. Šilhan, A. Taghavi-Chabert and V. Žádńık vanishing of the self-dual, respectively anti-self-dual part of the Weyl curvature of the induced conformal structure. In particular, the condition (3.8) in the previous proposition can be replaced by the condition that the conformal structure is half-flat. 4 Normalisation and characterisation By Proposition 3.8, for n ≥ 3, the induced conformal Cartan connection associated to a non-flat n-dimensional projective structure differs from the normal conformal Cartan connection for the induced conformal structure. In this section we will analyse the form of the difference and thus derive properties of the induced conformal structures. Furthermore, we will show that any split-signature conformal manifold having these properties is locally equivalent to the conformal structure on the Fefferman space over a projective manifold. 4.1 The normalisation process We are going to normalise the conformal Cartan connection ω̃ind ∈ Ω1 ( G̃, g̃ ) that is induced by a normal projective Cartan connection ω ∈ Ω1(G, g). Any other conformal Cartan connection ω̃′ differs from ω̃ind by some Ψ ∈ Ω1 ( G̃, g̃ ) so that ω̃′ = ω̃ind + Ψ. This Ψ must vanish on vertical fields and be P̃ -equivariant. The condition on ω̃′ to induce the same conformal structure on M̃ as ω̃ind is that Ψ has values in p̃ ⊆ g̃. One can therefore regard Ψ as a P̃ -equivariant function Ψ: G̃ → (g̃/p̃)∗ ⊗ p̃. According to the general theory as outlined in [11, Section 3.1.13] there is a unique such Ψ such that the curvature function κ̃′ of ω̃′ satisfies ∂̃∗κ̃′ = 0, and then ω̃′ is the normal conformal Cartan connection ω̃nor. The failure of ω̃ind to be normal is given by ∂̃∗κ̃ind : G̃ → (g̃/p̃)∗⊗ p̃. The normalisation of ω̃ind proceeds by homogeneity of (g̃/p̃)∗ ⊗ p̃, which decomposes into two homogeneous components according to the decomposition p̃ = g̃0 ⊕ p̃+. In the first step of normalisation one looks for a Ψ1 such that ω̃1 = ω̃ + Ψ1 has ∂̃∗κ̃1 taking values in the highest homogeneity, i.e., ∂̃∗κ̃1 : G̃ → (g̃/p̃)∗ ⊗ p̃+. To write down this first normalisation we employ Weyl structures G̃0↪→G̃. By Proposition 3.5 we can take a reduced Weyl structure, i.e., one that is induced by a reduction G0↪→G ↪→ G̃ with respect to the structure group Q0 := Q∩G0. This allows us to project ∂̃∗κ̃ind to ( ∂̃∗κ̃ind ) 0 : G0 → (g̃/p̃)∗ ⊗ g̃0 and to employ the G̃0-equivariant Kostant Laplacian �̃ : (g̃/p̃)∗ ⊗ g̃0 → (g̃/p̃)∗ ⊗ g̃0, �̃ := ∂̃ ◦ ∂̃∗+ ∂̃∗ ◦ ∂̃. For the first normalisation step we need to form a map Ψ1 : G̃ → (g̃/p̃)∗⊗ p̃ that agrees with −�̃−1 ( ∂̃∗κ̃ind ) 0 in the g̃0-component. If we have formed any such Ψ1 along G0↪→G̃ we can just equivariantly extend this to all of G̃. To proceed with the analysis of the normalisation we need to establish a couple of technical lemmas. As before, we denote by f◦ ⊂ p̃+ ∼= (g̃/p̃)∗ the annihilator of f = p/q ⊂ g/q ∼= g̃/p̃. Recall that f◦ = ϕ(p+) ∼= f [−2]. Lemma 4.1. Let V be a g-representation contained in a g̃-representation Ṽ and denote by φ 7→ φ̃ the inclusion Λkp+ ⊗ V ↪→ Λkp̃+ ⊗ Ṽ induced by ϕ : p+ → p̃+ and V ↪→ Ṽ . Then, for any φ ∈ Λkp+ ⊗ V , ∂̃∗φ− ∂̃∗φ̃ ∈ Λk−1f◦ ⊗ ( Λ2F̄ • V ) ⊆ Λk−1p̃+ ⊗ Ṽ . In particular, for the adjoint representations, ∂∗φ = 0 if and only if ∂̃∗φ̃ ∈ Λk−1f◦ ⊗ Λ2F̄ . Proof. For the sake of presentation, assume that φ is decomposable, i.e., of the form φ = Z1 ∧ · · · ∧Zk ⊗ v, where Zi ∈ p+ and v ∈ V . Let us denote by the same symbols also the images of these elements under the inclusion g ↪→ g̃ and V ↪→ Ṽ , i.e., Zi ∈ p̃ and v ∈ Ṽ , respectively. Let Z̃i ∈ f◦ be the images of Zi under the inclusion ϕ : p+ → p̃+. Now, by definition of the A Projective-to-Conformal Fefferman-Type Construction 17 Kostant co-differential, the difference ∂̃∗φ − ∂̃∗φ̃ evaluated on any k − 1 elements from g̃/p̃ is a linear combination of terms of the form( Zi − Z̃i ) • v. (4.1) However, the differences Zi− Z̃i ∈ p̃ are represented by the matrices as in (A.1) in the Appendix where only the Z-entries are non-vanishing and hence contained in Λ2F ∩ p̃ = Λ2F̄ . Thus (4.1) belong to the image of • : Λ2F̄ × V → Ṽ and the first claim follows. For the second claim we use that Λ2F̄ • g = [ Λ2F̄ , g ] ⊆ Λ2F and Λ2F ∩ g = 0: since ∂̃∗φ (evaluated on any k − 1 elements from g̃/p̃) has values in g ⊂ g̃, vanishing of ∂∗φ is equivalent to ∂̃∗φ̃ having values in Λ2F . But ∂̃∗φ̃ has generally values in p̃ and Λ2F ∩ p̃ = Λ2F̄ , hence the claim follows. � Lemma 4.2. If ψ ∈ p̃+ ∧ f◦ ⊗ Λ2F̄ ⊆ Λ2p̃+ ⊗ p̃ then ∂̃∗ψ ∈ f◦ ⊗ f◦ ⊆ p̃+ ⊗ p̃+. Proof. ψ is a sum of terms of the form Z1 ∧ Z2 ⊗ A, where Z1 ∈ p̃+, Z2 ∈ f◦ and A ∈ Λ2F̄ . Applying the Kostant co-differential gives ∂̃∗(Z1 ∧ Z2 ⊗A) = Z1 ⊗ [Z2, A]− Z2 ⊗ [Z1, A]. Now [Z2, A] belongs to [ f◦,Λ2F̄ ] = 0 and [Z1, A] belongs to [ p̃+,Λ 2F̄ ] = f◦, hence the claim follows. � The following lemma contains the crucial information which is necessary to perform our normalisation. We are going to specify the curvature function κ̃ind (later also κ̃nor) by describing its values along the natural Q-reduction G ↪→ G̃ over M̃ . Recall from Section 3.2 that Λ2F̄ is a Q-invariant subspace in g̃0, which can be identified with (Λ2f)[−2]. Lemma 4.3. For any u ∈ G, we have ∂̃∗κ̃ind(u) ∈ f◦ ⊗ Λ2F̄ ⊆ p̃+ ⊗ g̃0. Identifying Λ2F̄ ∼= ( Λ2f ) [−2] and f◦ ∼= f [−2], we have in fact ∂̃∗κ̃ind(u) ∈ ( f � Λ2f ) [−4], i.e., ∂̃∗κ̃ind(u) is contained in the kernel of the alternation map alt : ( f ⊗ Λ2f ) [−4]→ ( Λ3f ) [−4]. Proof. It is a general assumption that ω̃ind is induced by a normal projective Cartan connection on G, i.e., ∂∗κ(u) = 0, for any u ∈ G. Hence it follows from Lemma 4.1 that ∂̃∗κ̃ind(u) belongs to f◦ ⊗ Λ2F̄ ∼= ( f ⊗ Λ2f ) [−4]. Further we need a finer discussion involving the properties of κ : G → Λ2p+ ⊗ g to show that κ(u) belongs to the kernel of the Q-equivariant map Λ2p+ ⊗ g→ ( Λ3f ) [−4] given by φ 7→ alt ( ∂̃∗φ̃ ) . (4.2) Note that any element φ ∈ Λ2p+⊗ p+ for which ∂∗φ = 0 is mapped to zero: since φ̃ ∈ Λ2p̃+⊗ p̃ and [p̃+, p̃] = p̃+, the co-differential ∂̃∗φ̃ has values in f◦ ⊗ p̃+. But, by Lemma 4.1, it also has values in f◦ ⊗ Λ2F̄ and p̃+ ∩ Λ2F̄ = 0. Thus it suffices to consider the harmonic elements from Λ2p+⊗g0, i.e., the ones corresponding to the projective Weyl tensor. For that purpose we consider the simple part of Q0 = Q∩G0 which is isomorphic to SL(n−1), cf. the matrix realisation (A.2) in the Appendix where it corresponds to the A-block. Considering both Λ2p+ ⊗ g0 ∼= Λ2Rn∗ ⊗ Rn∗ ⊗ Rn and ( Λ3f ) [−4] ∼= Λ3Rn∗ as 18 M. Hammerl, K. Sagerschnig, J. Šilhan, A. Taghavi-Chabert and V. Žádńık representations of SL(n−1), the map (4.2) is either trivial or an isomorphism on each SL(n−1)- irreducible component. One can check that there is only one SL(n − 1)-irreducible component that occurs in both spaces, and it is isomorphic to Λ2Rn−1∗. Hence it suffices to compute (4.2) on one element contained in such component. Let Xn ∈ g− and Zn ∈ p+ be the two dual basis vectors stabilised by SL(n− 1) and consider an element φ = Z1 ∧ Z2 ⊗Xn ⊗ Zn − Z1 ∧ Z2 ⊗X1 ⊗ Z1 + Zn ∧ Z2 ⊗Xn ⊗ Z1. Indeed φ is completely trace-free, satisfies the algebraic Bianchi identity and the SL(n−1)-orbit of φ is isomorphic to Λ2Rn−1∗. Now, ∂̃∗φ̃ = −Z̃1 ⊗ Z̃n ∧ Z̃2 − Z̃n ⊗ Z̃1 ∧ Z̃2, which indeed lies in the kernel of the alternation map. Hence the statement follows. � We can now determine the form of the normal conformal Cartan connection: Proposition 4.4. The normal conformal Cartan connection is of the form ω̃nor = ω̃ind + Ψ1 + Ψ2, where Ψ1 = −1 2 ∂̃ ∗κ̃ind ∈ Ω1 hor ( G̃, p̃ ) and Ψ2 ∈ Ω1 hor ( G̃, p̃+ ) . Furthermore, along the reduction G ↪→ G̃ we have Ψ1 ∈ Ω1 hor ( G,Λ2F̄ ) , Ψ2 ∈ Ω1 hor(G, f◦). Remark 4.5. Since Ψ1 and Ψ2 are horizontal, they may equivalently be regarded as bundle- valued 1-forms on M̃ . Denoting by Λ2F̃ the associated bundle G ×Q Λ2F̄ over M̃ and by f̃◦ ⊆ T ∗M̃ the annihilator of f̃ = kerχ ⊆ TM̃ , Proposition 4.4 says Ψ1 ∈ Ω1 ( M̃,Λ2F̃ ) , Ψ2 ∈ Ω1 ( M̃, f̃◦ ) , Ψ1(v) = Ψ2(v) = 0, for all v ∈ Γ(kerχ). Below we also use the corresponding frame forms, i.e., the P̃ -equivariant functions φ1 : G̃ → (g̃/p̃)∗ ⊗ p̃ and φ2 : G̃ → (g̃/p̃)∗ ⊗ p̃+ such that, for any u ∈ G̃, Ψ1 = φ1(u) ◦ ω̃ind and Ψ2 = φ2(u) ◦ ω̃ind. In these terms, the proposition means that along the reduction G ↪→ G̃ these maps restrict to Q-equivariant functions φ1 : G → f◦ ⊗ Λ2F̄ , φ2 : G → f◦ ⊗ f◦. Further we put Ψ = Ψ1 + Ψ2 and φ = φ1 + φ2. Proof. The Kostant Laplacian �̃ restricts to an invertible endomorphism of ((g̃/p̃)∗⊗g̃0)∩im ∂̃∗ that acts by scalar multiplication on each of the G̃0-irreducible components. Now, restricting to G ↪→ G̃ and suppressing all arguments u ∈ G, it was shown in Lemma 4.3 that ∂̃∗κ̃ind is contained in one of the irreducible components, namely in ( f � Λ2f ) [−4]. On this component �̃ acts by multiplication by 2. Thus, the modification map accomplishing the first normalisation step is φ1 : G → f◦ ⊗ Λ2F̄ , φ1 := −1 2 ∂̃∗κ̃ind = −�̃−1∂̃∗κ̃ind. Now, let ω̃1 := ω̃ind + φ1 ◦ ω̃ind be the modified Cartan connection. The corresponding curvature function κ̃1 can be expressed in terms of κ̃ind, φ1 and its differential dφ1 so that κ̃1(X,Y ) = κ̃ind(X,Y ) + [ X,φ1(Y ) ] − [ Y, φ1(X) ] + dφ1(ξ)(Y )− dφ1(η)(X)− φ1([X,Y ]) + [ φ1(X), φ1(Y ) ] , (4.3) A Projective-to-Conformal Fefferman-Type Construction 19 where X,Y ∈ g and ξ = ( ω̃ind )−1 (X), η = ( ω̃ind )−1 (Y ), cf. [11, formula (3.1)]. For the last term we have [ φ1(X), φ1(Y ) ] = 0 since φ1(X) has values in Λ2F̄ . The first three terms are κ̃ind(X,Y ) + [ X,φ1(Y ) ] − [ Y, φ1(X) ] = κ̃ind(X,Y ) + ∂̃φ1(X,Y ), which by construction vanishes upon application of the Kostant co-differential, i.e., ∂̃∗ ( κ̃ind + ∂̃φ1 ) = 0. The remaining terms in (4.3) can be combined into a map Λ2g→ Λ2F̄ , (X,Y ) 7→ dφ1(ξ)(Y )− dφ1(η)(X)− φ1([X,Y ]), which vanishes upon insertion of two elements X,Y ∈ p. Therefore, applying Lemma 4.2, we conclude that ∂̃∗κ̃1 has values in f◦ ⊗ f◦. Thus the second modification map is φ2 : G → f◦ ⊗ f◦, φ2 := −�̃−1∂̃∗κ̃1. � 4.2 Properties The information provided in the previous proposition allows us to determine the properties satisfied by the normal conformal Cartan curvature: Proposition 4.6. The normal conformal Cartan curvature κ̃nor restricts to a map κ̃nor : G → Λ2(g̃/p̃)∗ ⊗ ( sl(n+ 1)⊕ Λ2F ) . (4.4) Moreover, the following integrability condition holds: iX κ̃ nor(u) ∈ f◦ ⊗ ( Λ2F̄ ⊕ f◦ ) , for all X ∈ f , u ∈ G. (4.5) Proof. Let κ̃nor be the curvature function of the normal Cartan connection ω̃nor = ω̃ind + φ ◦ ω̃ind, where φ = φ1 + φ2. With the same conventions as in the proof of Proposition 4.4, [11, formula (3.1)] yields κ̃nor(X,Y ) = κ̃ind(X,Y ) + [X,φ(Y )]− [Y, φ(X)] + dφ(ξ)(Y )− dφ(η)(X)− φ([X,Y ]) + [φ(X), φ(Y )]. Clearly, κ̃ind(X,Y ) has values in sl(n + 1) and vanishes upon insertion of X ∈ p. A term of the form [X,φ(Y )] vanishes if Y ∈ p and has values in [ p,Λ2F̄ ⊕ f◦ ] ⊆ Λ2F̄ ⊕ f◦ for X ∈ p. A term of the form dφ(ξ)(Y ) has values in Λ2F̄ ⊕f◦ and vanishes for Y ∈ p. The term φ([X,Y ]) has values in Λ2F̄ ⊕ f◦ and vanishes for X,Y ∈ p. The last term [φ(X), φ(Y )] vanishes for all X,Y ∈ g since φ(X) has values in Λ2F̄ ⊕ f◦. Altogether, we obtain (4.4) and (4.5). � We observe here that it follows directly from (4.4) that the pairing of κ̃nor with the involu- tion K vanishes, 〈κ̃nor,K〉 = 0. To derive properties of induced tractorial and underlying objects on the conformal structure we will need the following preparatory lemma. Lemma 4.7. Let V be a G̃-representation and v ∈ V an element which is stabilised under G ⊆ G̃. Let v ∈ Γ ( Ṽ ) be the section of the associated tractor bundle G̃ × P̃ V corresponding to the constant function G → V , u 7→ v, along G. Then the covariant derivative ∇̃norv corresponds to the Q-equivariant function G → f◦ ⊗ V, u 7→ φ1(u) • v + φ2(u) • v. 20 M. Hammerl, K. Sagerschnig, J. Šilhan, A. Taghavi-Chabert and V. Žádńık Proof. The covariant derivative ∇̃norv corresponds to the map X ∈ g 7→ ( ω̃nor )−1 (X) · v +X • v. (4.6) The first term in (4.6) vanishes since it is the directional derivative of the constant function v. Now ω̃nor = ω̃ind + φ1 + φ2, and since X • v = 0 the claim follows. � We now show that the distinguished tractors sE , sF and K on the Fefferman space are all given as BGG-splittings from their underlying objects. Moreover, several stronger properties hold: Proposition 4.8. Let sE ∈ Γ ( S̃± ) , sF ∈ Γ ( S̃− ) and K ∈ Γ ( AM̃ ) be the tractor spinors and the adjoint tractor, respectively, and let η = ΠS̃0 (sE), χ = ΠS̃0 (sF ) and k = ΠAM̃0 (K) be the corresponding underlying objects as in Proposition 3.2. (a) The tractor spinor sF is parallel, i.e., ∇̃norsF = 0. In particular, χ is a pure twistor spinor, sF = L S̃− 0 (χ) and Hol(c) ⊆ SL(n+ 1) n Λ2 ( Rn+1 )∗ ⊆ Spin(n+ 1, n+ 1). (b) The tractor spinor sE is a BGG-splitting, i.e., ∂̃∗ ( ∇̃norsE ) = 0 and sE = L S̃± 0 (η). (c) The adjoint tractor K is a BGG-splitting and k is a conformal Killing field, i.e., ∂̃∗∇̃norK = 0, K = LAM̃0 (k) and ∇̃norK = ikΩ̃ nor. Moreover, we have ikκ̃ nor = 2φΛ2F . (4.7) Proof. (a) Since φ1, φ2 have values in ker sF we have φ1 • sF + φ2 • sF = 0. Thus, according to Lemma 4.7, we have ∇̃norsF = 0 and the rest is obvious. (b) The spinor sE is of the form sE = ( ∗η ). According to Lemmas 4.3 and 4.7, φ1 has values in ( f � Λ2f ) [−4] and ∂̃∗ ( ∇̃norsE ) corresponds to ∂̃∗ ( φ1 • sE ) = (( φ1 • η ) Σ̃∓[− 1 2 ] 0 ) . The projection ( φ1 • η ) Σ̃∓[− 1 2 ] can be realised as the full (triple) Clifford action on φ1(u) ∈(⊗3 f ) [−4], where u ∈ G. Now it is easy to see that this action must vanish for a φ1(u) ∈( f � Λ2f ) [−4]: We realise φ1(u) equivalently in ( S2f ⊗ f ) [−4] by symmetrisation in the first two slots, then the complete Clifford action on η vanishes because the action of the first two slots is just a (trivial) trace multiplication. (c) According to Lemma 4.7, ∇̃norK corresponds to φ1 • K + φ2 • K. Since K/p̃ = k ∈ f , the previous element lies in p̃. In particular, ∇̃norK has trivial projecting slot, and thus k = Π0(K) is a conformal Killing field. Since φ2 •K ∈ p̃+, we have that ∂̃∗ ( ∇̃norK ) corresponds to ∂̃∗ ( φ1 • K ) . Now φ1 • K = −K • φ1 = 2φ1, since K acts by multiplication with −2 on Λ2F̄ . But φ1 ∈ im ∂̃∗ ⊆ ker ∂̃∗, and the expression ∂̃∗ ( φ1 • K ) therefore vanishes. The equality ∇̃norK = ikΩ̃ nor is just (2.13) for the conformal Killing field k with its BGG-splitting K. In terms of the Q-equivariant functions φ = φ1 + φ2 and κ̃nor along G ↪→ G̃, this can be expressed as φ •K = ikκ̃ nor, which yields (4.7). � We now collect the essential information about the induced conformal structure (M̃, c) which we derived: Proposition 4.9. Let ( M̃, c ) be the conformal spin structure induced from an oriented projective structure (M,p) via the Fefferman-type construction. Then the following properties are satisfied: A Projective-to-Conformal Fefferman-Type Construction 21 (a) ( M̃, c ) admits a nowhere-vanishing light-like conformal Killing field k such that the corre- sponding tractor endomorphism K = LAM̃0 (k) is an involution, i.e., K2 = idT̃ . (b) ( M̃, c ) admits a pure twistor spinor χ ∈ Γ ( Σ̃− [ 1 2 ]) with k ∈ Γ(kerχ) such that the corre- sponding parallel tractor spinor sF = L S̃− 0 (χ) is pure. (c) K acts by minus the identity on ker sF . (d) The following integrability condition holds: vawcW̃abcd = 0, for all v, w ∈ Γ(kerχ). (W) The only thing left to show for Proposition 4.9 is that the integrability condition (4.5) is equivalent to the condition (W) on the Weyl tensor: Lemma 4.10. Let ( M̃, c ) be a split-signature conformal spin structure endowed with trac- tors sE, sF and K satisfying conditions (a) and (b) from Proposition 4.9. Then condition (4.5) is equivalent to (W). Proof. The implication (4.5) =⇒ (W) is obvious. It remains to prove the converse implica- tion (W) =⇒ (4.5). By (W), one has that ( iX κ̃ nor ) g̃0 (u) ∈ ( f ⊗Λ2f ) [−4] ⊆ f◦⊗Λ2F̄ for X ∈ f , u ∈ G. Since sF is parallel with respect to ∇̃nor, we have κ̃nor(u) ∈ Λ2(g̃/p̃)∗ ⊗ ( p̃ ∩ ker sF ) . The projection of p̃∩ker sF to p̃+ is precisely f◦, hence it follows that ( iX κ̃ nor ) p̃+ (u) ∈ (g̃/p̃)∗⊗ f◦, and we obtain iX κ̃ nor(u) ∈ (g̃/p̃)∗ ⊗ ( Λ2F̄ ⊕ f◦ ) . We now prove that iX1iX2 κ̃ nor = 0 for all X1, X2 ∈ f . For this purpose it will be useful to work with the curvature form Ω̃nor, which we can represent as in (2.11). By (W) and the algebraic Bianchi identity, W̃abcd vanishes upon insertion of v, w ∈ Γ(kerχ) into any two slots, and in particular vawbW̃abcd = 0. Thus, it remains to check that vawbỸdab = 0. As in the proof of Proposition 3.2, a vector field w ∈ Γ(kerχ) corresponds to a section ( ∗ wd 0 ) ∈ Γ ( F̃ ) . According to (2.9), vaΩ̃nor ab •  ∗wd 0  = −vaỸrabwrvaW̃ c ab dw d 0  ∈ Γ ( Ẽb ⊗ T̃ ) . Since ivΩ̃ nor annihilates F̃ , it follows that vawrỸrab = 0. Using Ỹrab = −Ỹbra − Ỹabr, we obtain also vawbỸrab = 0. � 4.3 Characterisation We are now going to characterise the induced conformal structures. For this purpose we will introduce the following (’intermediate’) Cartan connection form: ω̃′ := ω̃nor − 1 2 ikκ̃ nor. (4.8) The following observation then follows immediately from Proposition 4.4 and formula (4.7): Lemma 4.11. The pullbacks of the Cartan connection forms ω̃′ ∈ Ω1 hor ( G̃, g̃ ) , ω̃nor ∈ Ω1 hor ( G̃, g̃ ) and ω̃ind ∈ Ω1 hor ( G̃, g̃ ) to G ↪→ G̃ agree modulo forms with values in p+ ⊆ sl(n+ 1) ⊆ g̃. 22 M. Hammerl, K. Sagerschnig, J. Šilhan, A. Taghavi-Chabert and V. Žádńık For the rest of this section, we will start with a given split-signature conformal spin struc- ture ( M̃, c ) satisfying all the properties of Proposition 4.9. In particular, M̃ is endowed with a conformal Killing field k ∈ Γ ( TM̃ ) , and we can still use formula (4.8) to define a Cartan connection ω̃′. The corresponding tractor connection will be denoted by ∇̃′ and the curvature by Ω̃′ or κ̃′. The following proposition now shows that the so constructed Cartan connection ω̃′ is in fact an SL(n+ 1)-connection. Proposition 4.12. Let ( M̃, c ) be a split-signature conformal (spin) structure satisfying all the properties of Proposition 4.9. Then the sections sF and K are parallel with respect to the tractor connection ∇̃′, i.e., ∇̃′sF = 0 and ∇̃′K = 0. In particular, Hol(ω̃′) ⊆ SL(n + 1) ⊆ Spin(n + 1, n + 1) and ω̃′ pulls back to a Cartan connection of type (SL(n+ 1), Q) with respect to the Q-reduction G ↪→ G̃. Along that reduction, the curvature functions κ̃′ and κ̃nor are related according to κ̃′ = ( κ̃nor ) sl(n+1) and κ̃′ satisfies the following integrability condition: iX κ̃ ′(u) ∈ f◦ ⊗ p+, for all X ∈ f, u ∈ G. (4.9) Proof. A tractor connection induced by ω̃′ can be written as ∇̃′ = ∇̃nor + Ψ with Ψ = −1 2 ikΩ̃ 1. That ∇̃′sF = 0 follows immediately from the fact that ∇̃′−∇̃nor = −1 2 ikΩ̃ nor has values in Λ2F̃ . Since k is a conformal Killing field we have ∇̃norK = ikΩ̃ nor. By definition ∇̃′K = ∇̃norK− 1 2 ikΩ̃ nor •K, which vanishes, since ikΩ̃ nor has values in Λ2F̃ and therefore 1 2 ikΩ̃ nor • K = ikΩ̃ nor. As in Proposition 4.9, we write the decomposition of T̃ into maximally isotropic eigenspaces of K with eigenvalues ±1 as Ẽ ⊕ F̃ . Since K is ∇̃′-parallel, it follows that this decomposition is preserved by ∇̃′. Moreover, since F̃ is the kernel of the pure tractor spinor sF it follows that Hol(ω̃′) ⊆ SL(n+ 1). In particular, ω̃′ reduces to a Cartan connection of type (SL(n+ 1), Q) on a Q-principal bundle G ⊆ G̃. We further compute that Ω̃′ = Ω̃nor − 1 2 d∇̃ nor ikΩ̃ nor = Ω̃nor − 1 2 d∇̃ nor∇̃norK = Ω̃nor − 1 2 Ω̃nor •K = Ω̃nor + 1 2 K • Ω̃nor = ( Ω̃nor ) (Ẽ⊗F̃)0 , where we are again using ∇̃norK = ikΩ̃ nor for the conformal Killing field k and that Ω̃nor has values in Ẽ ⊗ F̃ ⊕ Λ2F̃ . Stated for the corresponding curvature functions, this yields κ̃′ =( κ̃nor ) sl(n+1) . Moreover, since κ̃nor has values in ker sF ∩ p̃, it follows from (3.6) that (p̃ ∩ ker sF )sl(n+1) = p, thus κ̃′ has values in p. We know from (4.5) that iX κ̃ nor has values in Λ2F̄ ⊕ f◦ for X ∈ f . But since ( Λ2F̄ ) sl(n+1) = 0 and (p̃+ ∩ ker sF )sl(n+1) = p+, we obtain that (iX κ̃ nor)sl(n+1) has values in p+. Finally,( iX1iX2 κ̃ nor ) sl(n+1) = 0 for X1, X2 ∈ f follows immediately from (4.5), and altogether we ob- tain (4.9). � Next, before proving the main characterisation Theorem 4.14, we will show the following proposition on factorisations of particular Cartan geometries. This proposition can be under- stood as an adapted variant of [5, Theorem 2.7]. Proposition 4.13. Let ( G → M̃, ω ) be a Cartan geometry of type (SL(n+1), Q) with curvature κ : G → Λ2(g/q)∗ ⊗ g and let the following conditions be satisfied: iX1iX2κ(u) ∈ p, for all X1, X2 ∈ g/q, u ∈ G, A Projective-to-Conformal Fefferman-Type Construction 23 iX1iX2κ(u) ∈ p+, for all X1 ∈ p/q, X2 ∈ g/q, u ∈ G, iX1iX2κ(u) = 0, for all X1, X2 ∈ p/q, u ∈ G. Then G is locally a P -bundle over M = G/P and ω defines a canonical projective structure on M . Proof. The third of the above listed conditions implies that G is locally a P -bundle G → M by [5]. We will restrict G to assume this globally. We define M = G/P and G0 = G/P+. Let σ : G0 → G be a G0-equivariant splitting. It follows from the second of the above listed conditions that LζXω = −ad(X) ◦ ω mod p+, for all X ∈ p. Now define θ ∈ Ω1(G0, g−), γ ∈ Ω1(G0, g0) and ρ ∈ Ω1(G0, p+) via the decomposition σ∗ω = θ⊕γ⊕ρ. Since σ is G0-equivariant and the Lie derivative is compatible with pullbacks it follows that Lζ̄X (θ ⊕ γ) = −ad(X) ◦ (θ ⊕ γ), for all X ∈ g0. In particular, θ and γ are G0-equivariant and define a (reductive) Cartan geometry (G0 → M, θ ⊕ γ) of type (Rn o SL(n),SL(n)), i.e., an affine connection on M . Since by assumption Ω has values in p, θ ⊕ γ is torsion-free and so is the affine connection. Now take another splitting σ′ = σ · exp Υ, for some Υ: G → p+. Since Ad(exp Υ) acts by the identity on g− = g/p, one has (rexp Υ)∗ω = ω mod p, and thus θ is independent of the choice of splitting. Then σ′∗ω = θ⊕γ′⊕ρ′ and θ⊕γ′ = Ad(exp Υ)◦(θ⊕γ), projected to g−⊕g0. But since exp Υ ∈ P+, this shows that γ′ is projectively equivalent to γ. We thus obtain a well-defined projective structure on M . Since ω is P -torsion-free and P -equivariant modulo p+, it can be (uniquely) modified to a normal Cartan connection ωnor ∈ Ω1(G, g) with ωnor − ω ∈ Ω1(G, p+). In particular, each splitting σ : G0 → G is in fact a Weyl structure of the projective structure on M . � Theorem 4.14. A split-signature (n, n) conformal spin structure c on a manifold M̃ is (locally) induced by an n-dimensional projective structure via the Fefferman-type construction if and only if the following properties are satisfied: (a) ( M̃, c ) admits a nowhere-vanishing light-like conformal Killing field k such that the corre- sponding tractor endomorphism K = LAM̃0 (k) is an involution, i.e., K2 = idT̃ . (b) ( M̃, c ) admits a pure twistor spinor χ ∈ Γ ( Σ̃− [ 1 2 ]) with k ∈ Γ(kerχ) such that the corre- sponding parallel tractor spinor sF = L S̃− 0 (χ) is pure. (c) K acts by minus the identity on ker sF . (d) The following integrability condition holds: vawcW̃abcd = 0, for all v, w ∈ Γ(kerχ). Proof. Starting with a projective structure (M,p), it follows from Proposition 4.9 that the induced conformal structure ( M̃, c ) has all the stated properties. On the other hand, let ( M̃, c ) be a conformal structure with the stated properties. Then, by Proposition 4.12, ω̃′ restricts to a Q-equivariant Cartan connection form with values in sl(n + 1) on the reduction G ↪→ G̃. The corresponding curvature κ̃′ takes values in p and for X ∈ f we have that iX κ̃ ′ takes values in p+. It follows from Proposition 4.13 that ω̃′ factorises to a projective structure p on the leaf space M . 24 M. Hammerl, K. Sagerschnig, J. Šilhan, A. Taghavi-Chabert and V. Žádńık Let us now show that the two constructions are inverse to each other. Assume first that a conformal structure (M̃, c) is induced by a projective structure (M,p). Then according to Lemma 4.11 ω̃′ and ω̃ind agree modulo p+. This implies that the projective structure defined by ω̃′ is equal to the original projective structure. Conversely, assume now that (M,p) is a projective structure with associated Cartan geometry (G, ω′) that is induced from a conformal structure ( M̃, c ) with associated Cartan geometry ( G̃, ω̃nor ) . Since ω̃′ is not normal, but torsion- free, there is ϕ ∈ Ω1 hor(G, p+) such that ω′+ϕ is the normal projective Cartan connection. Since p+ ⊆ p̃ the induced conformal structure on M̃ agrees with the original conformal structure. Thus, the Fefferman-type construction (with normalisation) and the described factorisation are (locally) inverse to each other. � For a reformulation of the characterisation theorem in terms of underlying objects, see Sec- tion 6.2. 5 Reduced scales and explicit normalisation Although we obtained the desired characterisation in Theorem 4.14, we do not yet know the explicit relationship between the induced Cartan connection form ω̃ind and the normal conformal Cartan connection form ω̃nor. One of the aims of the present section is to obtain a formula for this difference, which is achieved in Theorem 5.7. As a consequence, we also obtain an explicit formula for the curvature Ω̃ind in terms of the normal conformal Cartan curvature Ω̃nor in Corollary 5.8. In this more refined analysis, reduced scales will play an important role. 5.1 Characterisation of reduced scales The notion of reduced Weyl structures and reduced scales is introduced in Section 3.4. Here we shall find an intrinsic characterisation (i.e., using the conformal structure only) of reduced scales and discuss their properties. As the scale bundle on the projective manifold M we may consider the positive elements in the density bundle E(1), which is the projecting part of the dual standard tractor bundle T ∗, see Section 2.4. Similarly, on the Fefferman space M̃ we take the positive elements in the density bundle Ẽ(1), the projecting part of the conformal standard tractor bundle T̃ . Hence for a projective scale ρ ∈ Γ(E+(1)) we have the tractor LT ∗ 0 (ρ) ∈ Γ(T ∗); similarly, for a conformal scale σ ∈ Γ ( Ẽ+(1) ) we have the tractor LT̃0 (σ) ∈ Γ ( T̃ ) . These will be termed scale tractors. On the one hand, reduced scales correspond to the sections of E+(1) → M seen as a subset of all sections of Ẽ+(1) → M̃ , see Remark 3.4. On the other hand, sections of T ∗ → M are understood as specific sections of the bundle F̃ → M̃ , which is a subbundle in T̃ → M̃ , see the generalities in Section 3.4 and the setup of our construction in Section 3.2. It follows that these two natural inclusions commute with the BGG-splitting operators. Lemma 5.1. The full arrows in the following diagram commute: Γ(T ∗) � � // Π0 �� Γ ( T̃ ) Π̃0 �� Γ(E+(1)) LT ∗ 0 EE � � // Γ ( Ẽ+[1] ) . LT̃0 XX Proof. Consider a projective density ρ ∈ Γ(E+(1)) on M , the corresponding tractor LT ∗ 0 (ρ) ∈ Γ(T ∗), and its extension to F̃ ⊆ T̃ , which is denoted by s′. The extension of ρ ∈ Γ(E+(1)) to Ẽ+[1] obviously coincides with the projection Π̃0(s′), and it is denoted by σ. We need to show A Projective-to-Conformal Fefferman-Type Construction 25 that s′ = LT̃0 (σ), i.e., that ∂̃∗∇̃nors′ = 0. According to Proposition 4.4, ω̃nor = ω̃ind + Ψ1 + Ψ2 with Ψ1 ∈ Ω1 ( M̃,Λ2F̃ ) , Ψ2 ∈ Ω1 ( M̃, f̃◦ ) , hence we have ∇̃nors′ = ∇̃inds′ + Ψ1 • s′ + Ψ2 • s′. Since Λ2F̃ acts trivially on F̃ ⊆ T̃ , we have Ψ1 • s′ = 0. Since f̃◦ ⊆ T ∗M̃ , it follows that ∂̃∗ ( Ψ2 • s′ ) = 0. It thus follows that ∂̃∗ ( ∇̃nors′ ) = ∂̃∗ ( ∇̃inds′ ) . Let φ be the frame form of ∇LT ∗0 (ρ). Then, according to Lemma 4.1, we have that ∂̃∗φ̃ = 0 since Λ2F • F = 0, and in particular ∂̃∗ ( ∇̃nors′ ) = 0. � We can now characterise reduced scales in terms of the corresponding scale tractors: Proposition 5.2. Let ( M̃, c ) be a conformal spin structure associated to an oriented projective structure (M,p) via the Fefferman-type construction. Let σ ∈ Γ ( Ẽ+[1] ) be a conformal scale and let s := LT̃0 (σ) ∈ Γ ( T̃ ) be the corresponding scale tractor. Then the following statements are equivalent: (a) The scale σ is reduced. (b) The tractor s is a section of F̃ ⊆ T̃ . (c) The twistor spinor χ is parallel with respect to the Levi-Civita connection D̃ of the metric corresponding to the scale σ. Furthermore, in a reduced scale, the Schouten tensor is strictly horizontal, i.e., it satisfies vaP̃ab = 0, for all va ∈ Γ(kerχ), (5.1) and the scalar curvature J̃ vanishes. Proof. (a) =⇒ (b): This follows from definitions and Lemma 5.1. (b) =⇒ (c): The condition (b) means that s · sF = 0. According to (2.6), (2.8) and (2.7), this condition expanded in slots yields s · sF = LT̃0 (σ) · LS̃0 (χ) = − 1 2n J̃σ 0 σ  ·( 1 n √ 2 D/ χ χ ) = ( − √ 2 2n J̃χσ − 1 nD/ χσ ) = ( 0 0 ) , where we use the Levi-Civita connection D̃ corresponding to σ. In particular, D/ χ = 0 and, since χ is a twistor spinor, the condition (c) follows. (c) =⇒ (b): The condition (c) yields sF = ( 0 χ ) according to (2.8). The fact that ∇̃asF = 0 yields P̃acγ cχ = 0 according to (2.7). Hence (5.1) holds, which in particular means J̃ = 0. Summarising, we have s = LT̃0 (σ) = 0 0 σ  and sF = LS̃0 (χ) = ( 0 χ ) . (5.2) Hence s · sF = 0 and the condition (b) follows. (b) and (c) =⇒ (a): According to the previous reasoning and (2.5), we have ∇̃nor a s =  0 σP̃ab 0  . Hence ∇̃nors is strictly horizontal, i.e., va∇̃nor a s = 0 for every va ∈ Γ(kerχ). Since ∇̃nor = ∇̃ind+Ψ and Ψ is horizontal, the horizontality of ∇̃nors is equivalent to the horizontality of ∇̃inds. Altogether, the condition (a) follows from Proposition 3.3 and Lemma 5.1. � 26 M. Hammerl, K. Sagerschnig, J. Šilhan, A. Taghavi-Chabert and V. Žádńık We will need some finer discussion on the slots of the distinguished tractor K =  ρa µab \|ϕ ka  ∈ Γ ( AM̃ ) (5.3) in reduced scales. From Proposition 4.8 we know that K is the BGG-splitting LAM̃0 (k), which in particular means that µab = D̃[akb] and ϕ = − 1 2nD̃ rkr according to (2.12). However, in the following statement we only exploit the algebraic properties of K, namely, that it acts by minus and plus the identity on F̃ and Ẽ , respectively. Lemma 5.3. Let us fix a reduced scale. Then the expression of K as in (5.3) satisfy ρa = 0, ϕ = −1, µa rvr = −va for every va ∈ Γ(kerχ) and µa rµrb = gab. Further we have µab = 〈γ[aγb]χ, η̄〉 for some η̄ ∈ Γ ( Σ̃∓ [ −1 2 ]) . Proof. Firstly, we use K • s = −s for any s ∈ Γ ( F̃ ) . The scale tractor s = LT̃0 (σ) of a reduced scale σ is a section of F̃ and it has the form as in (5.2). Thus it follows from (2.9) that ρa = 0 and ϕ = −1. Next, for every va ∈ Γ(kerχ), the tractor s = ( 0 va 0 ) is clearly a section of F̃ , since s · sF = 0. Thus it follows from (2.9) that µa rvr = −va. Secondly, we use K•s = s for any s ∈ Γ ( Ẽ ) . Considering the tractor s = ( 0 ωa 0 ) with arbitrary ωa ∈ Γ ( TM̃ ) , the tractor s̄ := s + K • s is a section of Ẽ , whose middle slot is ωa + µa rωr. It follows again from (2.9) that µa rµrb = gab. Thirdly, we use (3.3) which shows how K is built from sE and sF . Since the top slot of sF vanishes, middle slots of K are given by a suitable tensor product of χ and the top slot of sE . � We will also need more properties of conformal curvature quantities in reduced scales. Lemma 5.4. In a reduced scale, W̃abcdµ cd = 0, where µab = D̃[akb], (5.4) vcỸabc = 0, for all va ∈ Γ(kerχ). (5.5) Proof. In slots, the condition Ω̃nor ab • sF = 0 implies that W̃abcdγ cγdχ = 0. Pairing both sides of the latter equality with a spinor η̄ from Lemma 5.3 yields (5.4). Consider two arbitrary sections va, wb of f̃ = kerχ. Conditions (W) and (5.1) imply R̃abcdv awd = 0. Now, inserting va and we into the Bianchi identity D̃[aR̃bc]de = 0, we ob- tain vaweD̃aR̃bcde = 0, where we used the fact that f̃ is parallel. Since we can always regard gab as a section of f̃ ⊗ f̃∗, this implies 0 = gecvaD̃aR̃bcde = vaD̃aR̃icbc where R̃icbc is the Ricci tensor of D̃a. Since J̃ = 0, we have that P̃ab is proportional to R̃icab by a constant factor. Thus vaD̃aP̃bc = 0. From (5.1) and since f̃ is parallel we also have vbD̃aP̃bc = 0. Altogether, (5.5) follows by the definition of the Cotton tensor. � 5.2 Explicit normalisation formula So far we discussed three Cartan connections on the Fefferman space M̃ : the induced one ω̃ind (Section 3.3), the corresponding normal one ω̃nor (Section 4.1) and the modified auxiliary one ω̃′ (Section 4.3). Various properties of these and derived objects are enumerated in Propositions 4.9 and 4.12. The following proposition refines the integrability conditions included there. A Projective-to-Conformal Fefferman-Type Construction 27 Proposition 5.5. Let ( M̃, c ) be the conformal spin structure induced from an oriented projective structure (M,p) via the Fefferman-type construction. Then, along the reduction G ↪→ G̃, iX κ̃ nor(u) ∈ f◦ ⊗ Λ2F̄ , for all X ∈ f, u ∈ G, (5.6) iX κ̃ ′(u) = 0, for all X ∈ f, u ∈ G. (5.7) Proof. From (4.5) we already know that iX κ̃ nor has values in f◦⊗ ( Λ2F̄⊕f◦ ) . We note that the top slot of sections of Λ2F̃ vanishes in reduced scales, cf. (3.7). Thus the part in f◦ corresponds to vrỸabr for a v ∈ Γ(f̃), which however has to vanish by (5.5). Hence (5.6) follows. The last condition (5.7) follows from κ̃′ = ( κ̃nor ) sl(n+1) , cf. Proposition 4.12. � Since ω̃′ is an SL(n+ 1)-connection on G̃ → M̃ , it is the extension of a Cartan connection ω′, on G → M̃ . Now, due to (5.7), any section v ∈ Γ(kerχ) inserts trivially into its curvature. But this is the standard condition on the connection ω′ to be a Cartan connection also on the bundle G →M , i.e., to be a projective Cartan connection, cf. [5]. Furthermore, we will show that the descended Cartan connection is normal, i.e., ω′ = ω. To do this, we first compute ∂̃∗κ̃′ and then use the relation between the co-differentials ∂∗ on M and ∂̃∗ on M̃ discussed in Lemma 4.1. Proposition 5.6. The curvature κ̃′ satisfies ∂̃∗κ̃′(u) = ikκ̃ nor(u) ∈ f◦ ⊗ Λ2F̄ , for all u ∈ G. (5.8) Proof. We shall compute ∂̃∗Ω̃′ directly. First observe that using Proposition 4.12 we have Ω̃′ = ( Ω̃nor ) sl(n+1) = Ω̃nor + 1 2K • Ω̃nor. Hence ∂̃∗Ω̃′ = 1 2 ∂̃ ∗(K • Ω̃nor ) , since ∂̃∗Ω̃nor = 0. Using (5.3) and (2.11), we compute K • Ω̃nor ab as ρc µc0c1 \|ϕ kc  •  −Ỹdab W̃abd0d1 \| 0 0  =  ρrW̃abrc − µcrỸrab + ϕỸcab −2W̃ab r [c0µc1]r + 2k[c0 Ỹc1]ab \| krỸrab krW̃abrc  . In a reduced scale, from the previous display together with Lemmas 5.3 and 5.4 we compute ∂̃∗ ( K • Ω̃nor ab ) =  0 2krW̃rac0c1 \| 0 0  = 2krΩ̃nor ra , which yields (5.8). � Theorem 5.7. Let (G, ω) be a projective normal Cartan geometry over M and let ( G̃, ω̃ind ) be the conformal Cartan geometry over M̃ induced via the Fefferman-type construction. Then (a) ω̃ind = ω̃′ = ω̃nor − 1 2 ikκ̃ nor, (b) ω̃nor = ω̃ind + Ψ1, where Ψ1 = −1 2 ∂̃ ∗κ̃ind = 1 2 ikκ̃ nor. Proof. (a) We use that iX κ̃ ′ = 0 for all X ∈ f according to (5.7). Then Proposition 5.6 together with Lemma 4.1 imply that ∂∗κ′ = 0. Thus ω′ is projectively normal, and therefore we obtain ω̃′ = ω̃ind. (b) The normalisation process of Proposition 4.4 provides Ψ = Ψ1 + Ψ2 such that ω̃nor = ω̃ind + Ψ, where Ψ1, Ψ2 are the first and second normalisation steps. However since ω̃′ = ω̃ind, it follows from Proposition 5.6 and (4.8) that ∂̃∗κ̃′ = ∂̃∗κ̃ind is, up to a constant multiple, the difference between ω̃nor and ω̃ind. Therefore already the first normalisation step completes the normalisation, i.e., Ψ2 = 0. � 28 M. Hammerl, K. Sagerschnig, J. Šilhan, A. Taghavi-Chabert and V. Žádńık Using the explicit relationship provided in Theorem 5.7 we can also obtain a detailed descrip- tion of the difference between the induced and the normal Cartan curvatures: Corollary 5.8. In a reduced scale, we have the following relation between the curvatures of the induced and the normal conformal Cartan connection: Ω̃ind ab = Ω̃nor ab + 1 2 K • Ω̃nor ab =  −Ỹcab W̃abc0c1 − W̃ab r [c0µc1]r + k[c0 Ỹc1]ab \| 0 1 2k rW̃abrc  . (5.9) In particular, 1 2 ikW̃ is the torsion of the induced Cartan connection ω̃ind. Proof. We obtained the concrete expression of K • Ω̃nor in the proof of Proposition 5.6. Now Lemmas 5.3 and 5.4, and a short computation yields (5.9). � 6 Comparison with Patterson–Walker metrics and alternative characterisation In this section we will show that the Fefferman-type construction studied in this article is closely related to the construction of so-called Patterson–Walker metrics. These are the Riemann extensions of affine connected spaces, firstly described in [26]. A conformal version of this construction was obtained by [15] for dimension n = 2, and was treated by the authors of the present article in general dimension in [20]. 6.1 Comparison Let M be a smooth manifold and p : T ∗M → M its cotangent bundle. The vertical subbundle V ⊆ T (T ∗M) of this projection is canonically isomorphic to T ∗M . An affine connection D on M determines a complementary horizontal distribution H ⊆ T (T ∗M) that is isomorphic to TM via the tangent map of p. Definition 6.1. The Riemann extension or the Patterson–Walker metric associated to a tor- sion-free affine connection D on M is the pseudo-Riemannian metric g on T ∗M fully determined by the following conditions: (a) both V and H are isotropic with respect to g, (b) the value of g with one entry from V and another entry from H is given by the natural pairing between V ∼= T ∗M and H ∼= TM . It follows that V is parallel with respect to the Levi-Civita connection of the just constructed metric. Hence Patterson–Walker metrics are special cases of Walker metrics, i.e., metrics ad- mitting a parallel isotropic distribution. The explicit description of the metric g in terms of the Christoffel symbols of D can be found in [20, 26]. The previous definition can be adapted to weighted cotangent bundles T ∗M(w) = T ∗M ⊗ E(w), provided that M is oriented and D is special, i.e., preserving a volume form on M , which allows a trivialisation of E(w). It turns out that Patterson–Walker metrics induced by projectively equivalent connections are conformally equivalent if and only if w = 2 (interpreted as a projective weight according to the conventions from Section 2.4). Altogether, we have a natural split-signature conformal structure on T ∗M(2) induced by an oriented projective structure (M,p). From Section 3.3 we know that M̃ = T ∗M(2) \ {0} is the Fefferman space. Special affine connections from p are just the exact Weyl connections of the corresponding parabolic geometry. A Projective-to-Conformal Fefferman-Type Construction 29 The corresponding objects on M̃ are the reduced Weyl connections, respectively reduced scales, which correspond to distinguished metrics in the conformal class, see Section 3.4. We are going to show that these metrics are just the Patterson–Walker metrics. Proposition 6.2. Let ( M̃, c ) be the conformal structure of signature (n, n) associated to an n- dimensional projective structure (M,p) via the Fefferman-type construction. Then any metric in c corresponding to a reduced scale is a Patterson–Walker metric. Proof. Within the proof we refer to the notation and explicit matrix realisations from Ap- pendix A. By definition, the Fefferman space is M̃ = G/Q, which yields TM̃ ∼= G ×Q g/q. Conformally invariant objects on M̃ , respectively objects related to a choice of reduced scale, correspond to data on g/q ∼= g̃/p̃ that are invariant under the action of Q, respectively Gss0 ∩Q. Elements in g/q will be represented by matrices of the form− z 2 ∗ ∗ X ∗ ∗ w Y t − z 2  , where z, w ∈ R and X,Y ∈ Rn−1. Firstly, one verifies that Y tX − zw, (6.1) is the only quadratic form that is invariant under Gss0 ∩Q. Hence any reduced-scale metric in c corresponds to the quadratic form (6.1) in a suitable frame. Secondly, the vertical subbundle V ⊆ TM̃ corresponds to the Q-invariant subspace f = p/q ⊆ g/q given by X = 0 and w = 0. The horizontal distribution H ⊆ TM̃ induced by a linear connection from p corresponds to the unique (Gss0 ∩Q)-invariant subspace h ⊆ g/q complementary to f , which is given by Y = 0 and z = 0. Obviously, both f and h are isotropic with respect to (6.1). Hence any reduced-scale metric in c satisfies the condition (a) from Definition 6.1. Thirdly, the canonical identification V ∼= T ∗M(2) corresponds to an isomorphism f ∼= (g/p)∗(2) of Q-modules. Identifying (g/p)∗(2) with p+(2), it turns out to be given by− z 2 ∗ ∗ 0 ∗ ∗ 0 Y t − z 2  7→ 0 Y t −z 0 0 0 0 0 0  . Now, the inner product of any v ∈ f and u ∈ h coincides with the pairing of the correspond- ing elements v ∈ p+(2) and u ∈ g/p. Hence any reduced-scale metric in c satisfies also the condition (b) from Definition 6.1 and so it is a Patterson–Walker metric. � 6.2 Alternative characterisation We have characterised split-signature (n, n) conformal structures c on M̃ induced by an n- dimensional projective structure via the Fefferman-type construction in Theorem 4.14. Now we know these structures correspond to conformal Patterson–Walker metrics discussed in [20]. There we found the following characterisation in terms of underlying objects by direct computa- tions and spin calculus. Our aim here is to indicate how to reach the same result in the current framework. Theorem 6.3. A split-signature (n, n) conformal spin structure c on a manifold M̃ is (locally) induced by an n-dimensional projective structure via the Fefferman-type construction if and only if the following properties are satisfied: (a) ( M̃, c ) admits a nowhere-vanishing light-like conformal Killing field k. 30 M. Hammerl, K. Sagerschnig, J. Šilhan, A. Taghavi-Chabert and V. Žádńık (b) ( M̃, c ) admits a pure twistor spinor χ such that f̃ = kerχ is integrable and k ∈ Γ ( f̃ ) . (c) The Lie derivative of χ with respect to the conformal Killing field k is Lkχ = −1 2(n+ 1)χ. (d) The following integrability condition holds: vrwsW̃arbs = 0, for all vr, ws ∈ Γ(kerχ). (W) We now express the conditions from Proposition 4.9 in underlying terms: (i) For a conformal Killing field k with the splitting K as in (5.3), a straightforward compu- tation shows that the condition K2 = idT̃ is equivalent to kaka = 0, ρaρa = 0, µabk b = ϕka, µabρ b = −ϕρa, kaρa = ϕ2 − 1, µ c a µcb = gab + 2k(aρb). (6.2) (ii) For a twistor spinor χ, the corresponding tractor spinor sF = ( χ̄ χ ) ∈ Γ(S̃−) is parallel with respect to ∇̃nor. In particular, purity of sF can be checked at one point. If χ = 0, respectively χ̄ = 0, this tractor spinor is pure whenever χ̄, respectively χ, is pure. If χ 6= 0 and χ̄ 6= 0, the purity of ( χ̄ χ ) is equivalent to χ and χ̄ being pure and their kernels having (n− 1)-dimensional intersection, cf. [13, Proposition III-1.12] or [22, 30]. (iii) Let k be a conformal Killing field which splits to K and χ a twistor spinor which splits to sF . Then the condition Lkχ = −1 2(n + 1)χ is equivalent to K • sF = −1 2(n + 1)sF . If the tractor spinor sF is pure it has an (n+ 1)-dimensional maximally isotropic kernel ker sF . Then K• sF = −1 2(n+1)sF is equivalent to K acting by minus the identity on ker sF , which therefore coincides with the eigenspace of K corresponding to −1. The assumption on the pure twistor spinor χ in Theorem 6.3 guarantees the existence of a suitable compatible metric for which χ is parallel. This result is proved in [20, Proposition 4.2]. Henceforth we shall assume D̃χ = 0 where D̃ is the corresponding Levi-Civita connection. In particular, we have sF = L S̃− 0 (χ) = ( 0 χ ) , which is pure and parallel by observation (ii). Expanding the latter condition according to (2.7) yields vaP̃ab = 0, for all va ∈ Γ(kerχ), (6.3) For K = LAM̃0 (k), we know from observation (iii) that K acts by − id on ker sF . By the very same reasoning as in the first part of the proof of Lemma 5.3 it follows that ρa = 0, ϕ = −1, µa rvr = −va, for all va ∈ Γ(kerχ). (6.4) Now we are prepared to prove the theorem: Proof of Theorem 6.3. If (M̃, c) is induced by a projective structure, the stated properties hold according to Proposition 4.9 and previous observations. For the converse direction, it remains to show that K2 = idT̃ , which is equivalently characterised by the identities (6.2). Then the properties (a)–(d) of Proposition 4.9 will be satisfied and the result will follow from the characterisation Theorem 4.14. The expansion of the prolonged conformal Killing equation (2.13) for k gives D̃akb = µab + gab, (6.5) D̃aµbr = −2P̃a[bkr] −Wbrask s, (6.6) A Projective-to-Conformal Fefferman-Type Construction 31 according to (2.10) and (2.11). Next, from (6.4) we especially have µb rkr = −kb. Applying D̃a to both sides of this equality and using (6.5) we obtain( D̃aµbr ) kr + µb rµar + µba = −(µab + gab). From (6.6), (W) and (6.3) we have ( D̃aµbr ) kr = 0, hence the previous display shows µa rµrb = gab. This together with (6.4) implies that all identities from (6.2) are satisfied, hence K2 = idT̃ . � A Explicit matrix realisations Here we provide explicit realisations of the Lie algebras introduced in Section 3.2 in terms of matrices. We will consider the inner product h and the involution K on Rn+1,n+1 given by the block matrices h := ( 0 In+1 In+1 0 ) and K := ( In+1 0 0 −In+1 ) with respect to the standard basis (e1, . . . , e2n+2). Then E = 〈e1, . . . , en+1〉 and F = 〈en+2, . . . , e2n+2〉 and the decomposition (3.4) can be written as g̃ = Λ2(E ⊕ F ) = ( E ⊗ F Λ2E Λ2F E ⊗ F ) . For ṽ := e1 + e2n+2, the Lie algebra p̃ of the parabolic subgroup P̃ ⊆ G̃ is of the following form p̃ =  a U t w 0 −W t −b X B V W C −X 0 Y t c b Xt 0 0 −Y t −d −a −Xt 0 Y D −Z −U −Bt −Y d Zt 0 −w −V t −c  , (A.1) where a, b, c, d, w ∈ R with a − b = d − c, U, V,W,X, Y, Z ∈ Rn−1, B ∈ gl(n − 1) and C,D ∈ so(n− 1). The nilradical p̃+ = p̃⊥ is then of the form p̃+ =  a U t w 0 −V t −a 0 0 V V 0 0 0 0 a a 0 0 0 0 −a −a 0 0 0 0 −U −U 0 0 a U t 0 −w −V t −a  . A choice of Levi subalgebra g̃0 ⊆ p̃ determines a grading g̃ = g̃− ⊕ g̃0 ⊕ p̃+. We shall choose g̃0 = p̃ ∩ p̃op, where p̃op ⊆ g̃ is the stabiliser of the light-like vector en+2. Explicitly, g̃0 =  a 0 0 0 0 0 X B V 0 C −X 0 Y t c 0 Xt 0 0 −Y t −a− c −a −Xt 0 Y D −Z 0 −Bt −Y a+ c Zt 0 0 −V t −c  . 32 M. Hammerl, K. Sagerschnig, J. Šilhan, A. Taghavi-Chabert and V. Žádńık The embedding i′ : g ↪→ g̃ of Lie algebras has the form A 7→ ( A 0 0 −At ) . The subgroup Q = i−1(P̃ ) is contained in P , the stabiliser in G of v = (ṽ)E = e1; the inclusion of corresponding Lie algebras is q = g ∩ p̃ = a U t w 0 A V 0 0 −a  ⊆ a U t w 0 B V 0 Xt c  = p, where tr(A) = 0 and a+ tr(B) + c = 0. The standard projective grading g = g− ⊕ g0 ⊕ p+, g− =  0 0 0 X 0 0 y 0 0  , g0 = a 0 0 0 B V 0 Xt c  , p+ = 0 U t w 0 0 0 0 0 0  , is compatible with the previous conformal grading so that the reduced Lie subalgebra q0 := q∩g0 coincides with the intersection of g0 ∩ g̃0. Explicitly, q0 = a 0 0 0 A V 0 0 −a  , (A.2) where tr(A) = 0. Acknowledgements The authors express special thanks to Maciej Dunajski for motivating the study of this construc- tion and for a number of enlightening discussions on this and adjacent topics. 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A Projective-to-Conformal Fefferman-Type Construction

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